Our final module in this guide is hesi a2 math study guide, it deals with the various mathematical concepts you will need to know for the HESI A2 exam.
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Classification of numbers
These are the basic building blocks of mathematics so we must start by looking at critical terms that you should know and understand.
- Integer: This is any whole number that is positive or negative with zero included. There are no mixed numbers, fractions, or decimals when it comes to integers
- Prime number: A prime number is a whole number greater than 1. Multiplying two smaller natural numbers cannot produce it. It is divisible only by 1 and itself.
- Composite number: This includes any whole number more than 1 but it must have two or more factors. This means it is any whole number not considered a prime number. Take 8 for example, it has 1,2,4, and 8 as its factors.
- Even number: When divided by two, any number that doesn’t leave a remainder is considered an even number.
- Odd number: Any number that when divided by 2 leaves a remainder is considered an odd number.
- Decimal number: When part of the number that is less than one is shown, a number is said to be a decimal number, for example, 1.421.
- Decimal point: This separates the ones place from the tenths place in decimal numbers. It’s also used in currency.
- Decimal place: This is the position of a number found on the right-hand side of a decimal point. For example, take the number 1.312. 3 is the first place right of the decimal point, 1 is the second point and 2 is the third point.
- Rational numbers: All fractions, decimals, and integers are included in the term rational numbers. This also includes any repeating or terminating decimal numbers.
- Irrational numbers: Because they have an infinite number of decimal places, these cannot be written as either decimals or fractions. Within the number, there is no recurring pattern of digits either.
- Real numbers: These include a set of both irrational and rational numbers.
Used as a way to see the distance between numbers and the relationship between them, a number line is represented as a graph.
For example, it may have a point that reflects zero, with negative numbers to the left of it and positive numbers to the right.
Word form and place value of numbers
Understanding how a place value system works is a necessity when writing out numbers in word form or turning word form into numbers.
- 1,000: Place = thousands. Power of 10 = 103
- 100: Place = hundreds. Power of 10 = 102
- 10: Place = tens. Power of 10 = 101
- 1: Place = ones. Power of 10 = 100
- 0.1: Place = tenths. Power of 10 = 10-1
- 0.01: Place = hundredths. Power of 10 = 10-2
- 0.001: Place = thousandths. Power of 10 = 10-3
So let’s look at the number 5,657.09.
- 5: thousands
- 6: hundreds
- 5: tens
- 7: ones
- 0: tenths
- 9: hundredths
Understanding the concept of absolute value is essential when working with negative numbers.
When we talk of the absolute value of a number, we are referring to how far away it is from zero when represented on a number line.
This value is written as |x| where x is the value of the number.
Take 2, for example.
It would be written as |2| and that’s because there are two units in the distance between 0 and 2 on a number line.
The absolute value for -2 would be written as |-2|
A mathematical process that takes a certain value(s) as an input(s) and then produces an output from it is known as an operation.
Value operation value is the way in which elementary operations are often written out.
So let’s take the mathematical process of 1+2=3.
The values are 1 and 2, while the process in operation here is addition (noted by the + sign between the values).
When you carry out the operation, you get to the output, which is 3.
Here, the value of one quantity is increased by the value of another with both of these known as addends.
The result is known as the sum.
When it comes to addition, there are no restrictions put on the order that the values must take.
If the numbers have a sign before them such as – or +, there are some rules to follow.
If the signs are the same, add the absolute values of the addends and then the original sign can be applied to the sum.
- (+5) + (+7) = +12
- (-5) + (-7) = -12
You need to take the addends absolute values and subtract the smaller from the larger when the signs are different, however.
Following that, the larger value’s original sign is applied to the difference.
Here’s an example.
- (+5) + (-7) = -2
- (-5) + (+7) = +2
The opposite operation to addition is subtraction and here one quantity (the minuend) has its value decreased by another (the subtrahend).
- 5-2 = 3
In the above example, the result is 3 and the term this is known as the difference.
Unlike with addition, the order of the values does matter when it comes to subtraction, however.
When signed numbers are subtracted, the subtrahend’s sign must be changed but from that point the same rules we covered in addition apply.
Here’s an example:
- (+4) – (+8) = (+4) + (-8) = -4
The process of multiplication can be viewed as repeated addition.
Here, one number, known as the multiplier shows how many times another number, known as the multiplicand is to be added.
So if we take 3×2, it means 2+2+2=6.
When multiplying, the order is not a factor because the result, known as the product, will still be the same.
When multiplying signed numbers, the product will be positive if the signs are the same, for example, (+3) x (+3) = 9 and (-3) x (-3) – (-9).
The product will be negative, however, should the signs be opposite.
The product’s sign is determined by how many negative factors there are when two or more factors are multiplied with each other.
So the product is negative if there is an odd number of negative factors and positive if there is an even number of negative factors.
Here’s an example of that: (+4) x (-8) x (-2) = +64 and (-4) x (-8) x (-2) = -64.
The opposite operation of multiplication is division.
Here, one number, known as the divisor shows how many parts another number, known as the dividend should be divided into with the result referred to as the quotient.
So, if we look at 20 / 4 = 5, the divisor (20) is split into 4 equal parts with each part then equal to 5.
When carrying out this operation, the order of the numbers does affect the outcome.
The rules for dividing signed numbers are similar to those of multiplication.
The quotient will be positive if the dividend and the divisor have the same sign and negative when they have opposite signs.
When there are multiple operations taking place, parentheses will designate which of those operations should happen first.
So in the example 5-(3+2) = 0, we know we need to add 3+2 first, as they are in the parentheses.
This is a number (in superscript) that you will find next to another number at the top right-hand side of it.
This indicates the number of times the base number must be multiplied by itself and gives us a shorter way to write a long mathematical expression, for example, 23 = 2x2x2.
When a number has 2 as an exponent, it is “squared” while one with a 3 is “cubed”.
The term given to the exponent is “power” so 32 is said to be “3 to the 2nd power” or “ 3 raised to the power of 2”/
Another way of writing a fractional exponent is using a root that uses the radical symbol (√) to show the operation instead of using a superscript.
In a radical, you will find a number underneath the bar and in some cases, a number in the upper left as well, for example, 2√3.
Should no numbers appear on the left, the radical is a square root and this is the same number raised to the one-half power.
When a number has an integer for its square root, it is said to be a perfect square.
In the numbers from 1 to 100, you will find 10 perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.
In order to evaluate an expression accurately, you should follow a set of rules that tell us in which order the operations should be carried out.
In other words, it tells us which operation to do first when we have an expression that includes multiple different operations.
To remember this, you should think of the mnemonic PEMDAS or “Please Excuse My Dear Aunt Sally”.
This is broken down as follows:
- P: Parenthesis
- E: Exponents
- M: Multiplication
- D: Division
- A: Addition
- S: Subtraction
While one appears below the other, you should note that when it comes to multiplication and division, they have equal precedence.
This stands for addition and subtraction too.
In both these cases, the pairs are worked in an order from left to right.
So let’s look at an example of how PEMDAS would work using this: 5 + 20 / 4 x (2+3) -6.
- P: Operations in the parentheses must be carried out first, so (2+3)=5
- E: The exponents must be simplified, and so the equation now looks like this: 5 + 20 / 4 x 5-6
- MD: From left to right, perform the multiplication and division, so 20 / 4 = 5 and 5 x 5 = 25. This simplifies the equation to this: 5 + 25 – 6
- AS: From left to right, perform the addition and subtraction: 5 +25 = 30 then 30 – 6 = 24
Subtraction with regrouping
An excellent way to use the decimal systems’ built-in features include regrouping when carrying out operations that include long form subtraction.
On some occasions, the minuend is smaller than the subtrahend when subtracting within a place value.
In order to get a positive value, regrouping will allow you to borrow a unit from the place value to the left.
Mathematical symbols associated with word problems
You must be able to translate math words (which are verbal expressions) into math symbols when you are working with word problems.
Here’s a handy guide to various phrases and the math symbol associated with them.
- =: becomes, is the same as, costs, gets to, has, will be, was, is, equal
- x: triples, halves, doubles, twice, product of, multiplied by, of, time
- /: out of, ratio of, ratio to, per, divided by
- +: totals of, more than, and, combined, sum, added to, plus
- -: the difference between, minus, decreased by, less than, subtracted from
- X, n, etc: a variable, a number, how many, original value, how much, what
Factor and the greatest common factor
In order to obtain a product (or answer), factors are numbers that are multiplied together to do so.
So if we take the following equation, 3×3=9, we know that the product is 9, so therefore, the factors are 3 and 3.
While other numbers can have many factors, when it comes to a prime number, they can only have two, which are 1 and itself.
When a factor can divide exactly into two or more other numbers, it is termed a common factor.
Let’s take the factors of 12: they are 1, 2, 3, 4, 6, and 12 while the factors of 15 are 1,3, 5, and 15.
Therefore 12 and 15’s common factors are 1 and 3.
We also need to mention prime factors, which take the form of prime numbers.
In the example above, 2 and 3 are the prime factors of 12, while 3 and 5 are the prime factors of 15.
Next, we look at the greatest common factor (GCF).
The largest number that is a factor of two or more numbers is what this is.
So 1, 3, 5, and 15 are factors of 15, while 1, 5, 7, and 35 are factors of 35. In that example, 5 is then the GCF of 15 and 35.
Multiples and least common multiples
Multiples are sometimes seen listed in multiplication tables and a given factor’s integer increments.
Thus, an integer is a result of dividing a multiple by the factor.
So let’s take 7, for example.
The multiples of this are 1 x 7 = 7, 2 x 7 = 14, 3 x 7 =21, 4 x 7 = 28, 5 x 7 = 35.
The integers 1,2,3,4 and 5 are then a result of dividing 7,14,21, 28, or 35 by 7.
The smallest number that is a multiple of two (or more) numbers is called the least common multiple (LCM).
So let’s take 3.
The multiples thereof are 3,6,9,12,15 and so on while if we take 5, its multiples are 5,10,15,20, and so on.
The least common multiple of 3 and 5 is therefore 15.
When a number is expressed as one integer written above another, it is called a fraction.
It will include a dividing line between the numbers.
Fractions represent the quotient of the two numbers, for example, in this fraction (½), it’s 1 divided by 2.
In this example, 1 is called the numerator, while 2 is the denominator.
The number of parts under consideration is represented by the numerator while the total number of equal parts is represented by the denominator.
So the 2 in ½ is of two equal parts that the whole consists of.
Zero can never be the denominator of a fraction.
By multiplying or dividing (but not adding or subtracting), both the numerator and denominator by the same number, fractions can be manipulated, without their value changing.
You will simplify or reduce a fraction when dividing both numbers by a common factor.
If fractions are expressed differently, but still have the same value, the term applied to them is equivalent fractions.
When finding a common denominator in fractions, they are being manipulated so that they have the same denominator.
If this is done, the common denominator chosen should be a number that is the least common multiple of the two original denominators.
Proper fractions and mixed numbers
A proper fraction is one where the denominator is larger than the numerator.
If the opposite is true, the fraction is said to be an improper fraction.
Values of greater than one are associated with improper fractions, while those less than one are associated with proper fractions.
A number that includes both a fraction and an integer is known as a mixed number and improper fractions can be rewritten in this manner.
In much the same way, a mixed number can also be rewritten as an improper fraction.
The addition and subtraction of fractions
Fractions can be added and subtracted if they have the same common denominator.
This is achieved by adding or subtracting the two numerators while retaining the same denominator.
Should the denominator on the two fractions not be the same, they will need to be manipulated so that a common denominator is achieved before they can be added or subtracted.
Multiplying and dividing fractions
By multiplying the two numerators to obtain the new numerator and doing the same for their denominators, it’s simple for two fractions to be multiplied.
When dividing two fractions, the numerator and the denominator of the second fraction are flipped and from there, treat it as if it were multiplication.
Multiplying by a whole number, decimal, or fixed number
Usually you should convert to an improper fraction first when you are multiplying a mixed number by something.
Also, should the multiplicand be a decimal, it is easiest to first convert it to a fraction.
Let’s also just recap on what a mixed number and a decimal are.
- Mixed number: A fraction that’s added to a whole number
- Decimal: A sum of fractions representation
In the next step, the added quantities must be expressed with the same denominator, and to do this, you would multiply or divide the whole number by the fraction’s denominator.
You can now add the fractions when they have the same denominators and then perform the last multiplication before simplifying.
The parts of a whole can be represented using decimals.
Each number to the decimal point’s right shows the number of units of a corresponding power of ten when using the place value system.
Put the decimal in the numerator with 1 as the denominator if you would like to write a decimal as a fraction.
Now multiply by tens, both the numerator and denominator until no decimal places remain.
From here, the fraction must be simplified to its lowest terms.
Operations with decimals: Add and subtract
The decimal points must always be aligned if you are going to add or subtract decimals and the process is much like adding whole numbers.
Here’s an example:
- 1.5 + 1.5 = 3.0
By aligning all the decimal points visually in a vertical column, you have an easy way to add decimals.
This helps determine the final answer by showing exactly where the decimal points should be and one should always start with the addition from right to left.
If a column adds up to more than 9, the number should be carried to the left.
You can use the same rule when carrying out decimal subtraction.
Operations with decimals: Multiplication
We know that the two components of any multiplication problem are the multiplicand and the multiplier.
In decimal multiplication, it’s easier to look at the numbers not as decimals but rather, as whole numbers.
When you have calculated the final product, you should look at both the multiplicand and the multiplier and coin the numbers to the right of both.
The next step is to count the number of places to the product’s right and the decimal is then placed in that position.
Here’s an example.
- 12.3 x 2.56 has three places to the right of their decimals. Remove the decimal and multiply 123 x 256 which is 31,488. Starting on the right, count three places to the left and that’s where the decimal is inserted. So 31.488 is the, therefore, product.
Operations with decimals: Division
A divisor and dividend are found in every division problem.
That number that is being divided is the dividend while the number that the dividend is being divided by is the divisor.
In this example, 12/6, 12 is the dividend, and 6 is the divisor.
When there are decimals in division problems, you should always convert the divisor into a whole number.
To do this, move the divisor’s decimal to the right until the point that a whole number is formed.
After that, the decimal in the dividend should be moved to the right by the same number of spaces.
So in the example, 4.9 divided into 24.5, this would become 49 divided into 245 if you follow the rules above.
However, to create a whole number in the divisor, the decimal only needed to move one space to the right.
The problem can be carried out normally when the whole number is created: 245/9 = 5.
Percentages can be seen as fractions based on a whole of 100.
In other words, as one whole equalling 100%.
You’ll find three ways in which percentage problems are often presented:
- Calculate the percentage a number is of another number, for example, what percentage of 20 is 5?
- Calculate what number is a given percentage of a provided number, for example, what number is 10% of 40?
- Calculate what number another number is a given percentage of, for example, What number is 5 20% of?
In each of these examples above, there are three specific elements: the whole (W), the part (P), and the percentage (%).
These are used in the equation: P = W x % which can be rearranged in various ways should it be needed as a way to suit the question better.
When you see percentage problems, they are often presented in a word format, and because of this, working out which quantities are which will go a long way to help solve them.
Converting between decimals, fractions, and percentages
By moving the decimal point, you can convert decimals to percentages and vice versa.
- By moving the decimal point two places to the right, you can convert from a decimal to a percentage
- By moving the decimal point two places to the left, you can convert from a percentage to a decimal
In doing so, it’s helpful to remember that when compared with the equivalent decimal number, the percentage number is always larger.
In converting a fraction to a decimal, you need to divide the numerator by the denominator.
In converting a decimal to a fraction, you need to put the decimal in the numerator while the denominator is 1.
The numerator and denominator are then multiplied by tens until the decimal place is no more after which the fraction is simplified to its lowest terms.
By finding an equivalent fraction with a denominator of 100, fractions can be converted to a percentage.
To do this you first need to divide the percentage number by 100.
Follow that the fraction must be reduced to its simplest possible terms.
When we talk of the term rational, it refers to the fact that it is as a ratio or fraction that a number can be expressed.
Ratios and proportions
When two quantities have a relationship in which changes in one dictate how the other will change, it is said to be a proportion.
The way in which quantity might increase in a relationship by a set amount after an increase or decrease in another quantity is said to be a direct proportion.
An example of this would be the overall time of a car trip increasing as the trip distance increases.
There is a directly proportional relationship between the distance that needs to be traveled and the time required for that travel.
When one quantity increases and the other decreases, the relationship between them is said to be an inverse proportion relationship.
So if a car increases its speed, the time it will take to reach its destination will decrease.
In this case, the time is inversely proportional to speed.
When two quantities are compared in a particular order, that comparison is a ratio.
So if we take 14 cans of soda at a party and there are 20 kids, the soda-to-child ratio is 20 to 14.
This would be more commonly written as 20:14.
By dividing both sides by two, a ratio is usually reduced to its smallest whole number representation.
So 20:14 becomes 10:7.
Quantities exist in a constant of proportionality when there is a proportional relationship between them.
Here, the product of this constant as well as one of the quantities is equal to the other.
Let’s look at an example.
An apple costs $0.10, two apples cost $0.20, and three apples cost $0.30.
The number of apples and their total cost has a proportional relationship between them.
The contrast of that proportionality is the unit price of the apples, which is $0.10.
There’s an equation here too, specifically when it comes to finding the total price of the apples.
That variable (t) is worked out by multiplying the apples’ unit price (p) by their number (n), and therefore, the equation is: t=pn.
Unit rate expresses something’s quantity in terms of one unit of something else.
Let’s explain this a little better.
Let’s say you are traveling on an electric scooter and you cover 20 miles every two hours.
The unit rate here would express this as one hour, so in one hour you travel 10 miles and that’s your unit rate.
As a way to help solve problems, we use unit rates to compare different situations.
They have another application, however, and that’s the fact that they can be used to work out the amount of time an event will go on for.
So let’s say if you paint two equal-sized fences in two hours, the unit rate is 1. Know that, you can multiply it by however many equal-sized fences you want to paint to see how long it will take you.
When working with two points on a graph, we can use a specific equation to work out the slope between them using this formula;
- m = (y₂ – y₁) / (x₂ – x₁)
In this formula, (y₂ – y₁) and (x₂ – x₁) are the two points and all you have to do is simply plug in their numbers and solve the equation.
A negative slow will show a downward direction from left to right.
Terms and coefficients
A combination of one or more values that are arranged and added to each other are called mathematical expressions.
So, in theory, expressions could therefore be single numbers, but zero would also fall into this category.
The product of a real number is known as a variable term or coefficient.
Constants, sometimes called constant terms, can also be expressions that can include numbers that don’t have variables.
Here’s an example of what we mean.
Take the expression 6s2.
This is a single term with a real number (6) which is the coefficient.
The variable term is s2.
The sum of a single variable term in which there is no exponent for the variable, and a constant (that can be zero) is known as a single variable linear expression.
Here’s an example.
Consider the expression 2w + 7.
Here, 7 is the constant term while 2w is the variable term.
It’s critical to note here that a + or – sign are what separates terms.
Expressions are sums of terms and for that reason, one such as 6x-4 can be written as 6x + (-3).
When using this format, an emphasis is placed on the fact that the constant term is negative.
The perimeter of a square, often shown as 4s, is an example in the real world of a single variable linear expression.
For the most part, linear terms are the sum of any number of variable terms.
None of them, however, can include an exponent.
So using that rule, 4y2 is not a linear expression, while 2m + 7n – ¼p + 4.5q -1 is.
Here’s another example.
The expression of the area of a square, in which the side lengths are squared, is not a linear expression, while the sum of the sides of a triangle, which determines its perimeter, is.
A one variable linear equation is one which can be written as a + b = 0 (where a ≠ 0).
The root is the solution to these equations.
In the case of an equation like this, 5x + 10 = 0, when solving for x, you will find that the answer (or the root thereof) is -2.
Solving this starts with looking at both sides and subtracting 10 from them which then gives 5x = -10.
Then take the coefficient of the variable and divide both sides by it, in this case, that’s 6 and it gives you an answer of x=-2.
By plugging this back into the equation you started with, you can double check that you have the right answer, so that’s (5) (-2) = 10 = -10 + 10 = 0.
The set of all solutions to an equation is known as the solution set so in the example above, there is only one solution and that’s -2.
Should there have been more solutions that could have worked, they would have formed part of the solution set with the answer of -2.
In some cases, equations might have an empty set.
You can probably guess what that is from the name.
Simply put, it’s when there is no true solution to an equation.
If there are identical solution sets to more than one equation, they are said to be equivalent equations.
There are many ways in which linear equations can be written and this includes:
- Standard form
- Slope intercept form
- Point-slope form
- Two-point form
- Intercept form
You will find examples of all of these types in your coursework.
Solving linear equations that have one variable
When starting, as a way to eliminate any fractions, multiple terms must be multiplied by their lowest common denominator.
Next, as a means to isolate variables on one side of the equal sign, you can look from addition and subtraction to undo.
Then you take the coefficient of the variable and divide both sides by it to give you the value of the said variable.
Once you have that, this value can be substituted in the original equation, and in doing so, you can make sure you have the true equation.
Rules: Manipulating equations
When manipulating equations there are some rules that you need to follow.
When terms in an equation have the same variable, they are called like terms.
Whether they have the same coefficient or not does not play any role at all.
Terms included in this are even those that don’t have a variable at all as well as all constants.
The term that has a variable raised to the same power is a like term in equations that have terms with variables raised to different powers.
Let’s look at this example: x2 + 3x +2 = 2x2 +x -7 +2x.
- The like terms are 2 and -7 and they are both constants
- Because they include the variable x raised to the first power, as per the rule covered above, 3x, x, and 2x are like terms
- Because they include the variable x raised to the second power, x2 and 2x2 are also like terms.
- Note, however, that 2x2 and 2x are not like terms even though they both include the x variable and have the same coefficient
Ensuring that the same operation is carried out on both sides of the equation
A general rule of thumb is to ensure that any operations are carried out on both sides of an equation when trying to solve it.
In doing so, operations that will simplify the equation should be carried out.
This is important because the meaning of the equation is left unchanged when carrying out the same operations on both sides.
So what’s left when you carry this out is then the equivalent of the original equation.
If you didn’t carry out the operations on both sides, this would not be the case.
The benefit of combining like terms
When like terms are combined, all it means is that they are added or subtracted when they have the same variable.
By doing so, a set of like terms can be reduced to a single term, and therefore, the equation is simplified.
More often than not, as a first step in solving an equation, like terms can be combined.
Let’s look at an example: 2 (x + 3), +3(2 +x + 3) = -4.
- In the second set of parenthesis, the 2 and 3 are like terms
- If they are combined, we have 2(x + 3) +3 (x + 5) = -4
- The next step is to do the multiplication that the parentheses imply while distributing the outer 2 and 3 appropriately: 2x + 6 + 3x +15 = -4. Add 2x and 3 x together because they are like terms which leaves us with 5x +6 +15 = -4.
- Because they are like terms, constants 6, 15, and -4 can be combined, so 6 and 15 can be subtracted from both sides of the equation which leaves 5x – 4-6-15 which is 5x = 25 and therefore when simplified in the final step, x=5
Terms on opposite sides of the equation canceled
If they match each other, terms on opposite sides of the equation can be canceled.
This match, however, must be exact and that means that not only must they have the same variable but that it must be raised to the same power.
They must also have the same coefficient.
Let’s look at the following example: 3x + 2x2 +6 = 2x2 -6.
Here, it’s on both sides of the equation that 2x2 appears and because of this, it can be canceled.
This leaves the equation at 3x +6 = -6.
Don’t be fooled by the 6 on either side of the equation.
That cannot be canceled because one side is positive and the other negative.
They can be combined, however, as they are like terms which then leave the equation as 3x = -12, and if you simplify that further, x = -4.
One final thing to note when canceling terms is that they cannot be part of a larger term, and must be independent themselves.
Manipulating equation processes
Isolating variables is a critical part of the process of manipulating equations as by doing so, it’s on one side of the equation that the variable will then appear.
Equations are solved when on one side of it, you have an isolated variable, and on the other, it has been simplified as far as it can.
Only one variable will need to be isolated when dealing with a two-variable equation.
In an equation where the variable only seems raised to the first power – or linear equations, to isolate the variable, you would need to move all the terms that contain the variable to one side.
The other side will then comprise the remaining terms.
When we move a term, all we are doing is adding the inverse thereof to both sides.
Note that moved terms need to have their signs flipped when going to the other side of the equation.
After this has been done, the like terms on both sides of the equation are then combined and the final step is to find the coefficient of the variable, if applicable and divide both sides by it.
Equations having more than one solution
Non-linear equations can have more than one solution.
Here’s an example: x2 = 4 has a solution of both 2 and -2.
Another example of equations that have more than one solution are those that have absolute values, for example, |x| = 1 could see ex being 1 or -1.
Some linear equations too can have more than one solution.
For this to be the case, however, the equation must be true, irrespective of the variable’s value.
When this happens, there are infinitely many possible solutions that can be considered.
Equations that have no solution
Equations involving squares of variables and other types of nonlinear equations can have no solution.
Let’s look at the example of x2 = -2.
Because the square of any real number is always a positive, this has no solution in the real numbers.
Here’s another example, |x| = -1.
There is no solution here because the number’s absolute value is always positive.
Other equations that can have no solutions include those that don’t have powers greater than one, absolute values, or additional special functions.
Here’s another example of an equation that has no solution: 2 (x + 3) + x = 3x.
To try to solve it, you would distribute, which leaves 2x + 6 + x = 3x.
You’ll find that when the terms and variables are combined, they cancel each other, which leaves just 3x on both the left and right which is 6 = 0, a false answer.
Units of measurement
Prefixes: Metric measurement
- Giga: One billion, for example, one billion watts = 1 gigawatt
- Mega: One million, for example, one million hertz = 1 megahertz
- Kilo: One thousand, for example, one thousand grams = 1 kilogram
- Deci: One-tenth, for example, one-tenth of a meter = 1 decimeter
- Centi: One-hundredth, for example, one-hundredth of a meter = 1 centimeter
- Milli: One-thousandth, for example, one-thousandth of a liter = 1 milliliter
- Micro: One-millionth, for example, one-millionth of a gram = 1 microgram
The goal of converting between units is to ensure that the meaning stays the same, even though the way in which it will be displayed differs.
When moving from a larger to a smaller unit, the number of the known amount should be multiplied by the number of the equivalent amount.
Should you want to move from smaller to larger units, the number of the known amount should be divided by the number of the equivalent amount.
Conversion fractions are helpful when dealing with complicated conversions.
With these, the conversion factor is one of the fractions with the other having a numerator of an unknown amount.
It’s therefore in the denominator that the unknown amount is placed.
In some cases, the second fraction has the known value.
This comes from the problem in the numerator, with the unknown in the denominator.
To get the converted measurement, you can multiply the two fractions.
Note that the value of the fraction will be 1 because its numerator and denominator are equal.
Because of this, even though they have different numbers, the result in new units is equal to the result of old units.
Typical units and equivalents
- 1000 μg (microgram) = 1mg
- 1000 mg (milligram) = 1 g
- 1000 g (gram) = 1 kg
- 1000 kg (kilogram) = 1 metric ton
- 1000 mL (milliliter) = 1 L
- 1000 μm (micrometer) = 1 mm
- 1000 mm (millimeter) = 1 m
- 100 cm (centimeter) = 1 m
- 1000 m (meter) = 1 km
Distance and area measurement
- Inch (in): 1 inch (US equivalent) | 2.54 centimeters (metric equivalent)
- Foot (ft): 12 inches (US equivalent) | 0.305 meters (metric equivalent)
- Yard (yd): 3 feet (US equivalent) | 0.914 meters (metric equivalent)
- Mile (mi): 5280 feet (US equivalent) | 1.609 kilometers (metric equivalent)
- Acre (ac): 4840 square yards (US equivalent) | 0.405 hectares(metric equivalent)
- Square mile (sq. mi or mi.2: 640 acres (US equivalent) | 2.590 square kilometers (metric equivalent)
- Fluid ounce (fl oz): 8 fluid drams (US equivalent) | 29.573 milliliters (metric equivalent)
- Cup (c): 8 fluid ounces (US equivalent) | 0.237 liters (metric equivalent)
- Pint (pt): 16 fluid ounces (US equivalent) | 0.473 liters (metric equivalent)
- Quart (qt): 2 pints (US equivalent) | 0.946 liters (metric equivalent)
- Gallon (gal): 2 quarts (US equivalent) | 3.785 liters (metric equivalent)
- Teaspoon (t or tsp): 1 fluid dram (US equivalent) | 5 milliliters (metric equivalent)
- Tablespoon (T or tbsp): 4 fluid drams (US equivalent) | 15 milliliters (metric equivalent)
- Cubic centimeter (cc or cm3): 0.271 dram (US equivalent) | 1 milliliter (metric equivalent)
- Ounce (oz): 16 drams (US equivalent) | 28.35 grams(metric equivalent)
- Pound (lb): 16 ounces (US equivalent) | 453.6 grams (metric equivalent)
- Ton (tn): 2000 pounds (US equivalent) | 907.2 kilograms (metric equivalent)
Clarifications: Volume and weight measurements
We’ve covered various volume and weight measurements above, but, when using ounces and fluid ounces, you should take care, as they are not equivalent.
- 1 pint = 16 fluid ounces but 1 fluid ounce ≠ 1 ounce
- 1 pound = 16 ounces but 1 point ≠ to 1 pound
Also, not the word “ton” which in the US is 2000 pounds which is not the same as a metric ton of 1000 kilograms.
Time: Military format
The most popular time system that you will come across is sometimes called military time, but you will recognize it as the 24-hour clock.
While minutes and seconds are the same as you’d find using the 12-hour clock formation, the difference is that four figures are used to express time.
Time runs from 0000 hours (12 am on a 12-hour clock) to 2359 (11.59 pm),
While these are not used much in our modern world, you will need to know the basics about them as it could come up in the exam.
Here’s a breakdown that you can understand easily:
- Roman numeral I has a value of 1
- Roman numeral V has a value of 5
- Roman numeral X has a value of 10
- Roman numeral L has a value of 50
- Roman numeral C has a value of 100
- Roman numeral D has a value of 500
- Roman numeral M has a value of 1000
Placed together, the values of roman numerals are usually added, so XV1 = 16 (10 + 5 +1).
Note, however, that if the smaller number is placed first (in front of the larger number), then it must be subtracted from it, so IX = 9 (10-1).
There you have it for our HESI A2 study guide.
We hope you find the information in it useful.
Always remember, however, that this is a guide and does not supplement the coursework in any way.