High School Algebra I Unlocked (2016)
Chapter 4. Equations and Inequalities
Lesson 4.2. Single-Variable Inequalities
Game time! It’s a Tuesday night and you’ve decided to play a board game. You notice that the box says the game is for “ages 12 and over.” In other words, you can be 12, 24, or even 110 years old to play the game, but not 11, 4, or 0. This is a real-world example of an inequality, which could also be expressed as “necessary age ≥ 12,” which means the necessary age to play the game is greater than or equal to 12.
Unlike equations, which allow you to find the exact value of a variable, inequalities have a range of values for a variable. There are four signs you should be familiar with when working with inequalities:
Symbol |
Meaning |
> |
greater than |
< |
less than |
≥ |
greater than or equal to |
≤ |
less than or equal to |
How do these inequality symbols work? Well, if you’re told that x > 10, you know that x is greater than 10. Therefore, x can be any number that’s greater than 10, but x cannot be equal to 10 or any number that’s less than 10; e.g., xcould equal 11, 110, 1,110, or 1,100,000,000, but not 10, 0, −10, −1,100, or −1,100,000,000. In short, an inequality defines a range of values for a variable without giving you one specific value.
Here is how you may see single-variable equations on the SAT.
If = 2, what is the value of x ?
A) 4
B) 16
C) 16
D) 32
Solving Single-Variable Inequalities
Inequalities can be solved in the exact same way as equations, with one minor difference: If you multiply or divide by a negative number, you must flip the orientation of the inequality sign.
Let’s take a look at an example.
EXAMPLE
If 4x + 12 > 16, what is the expression that gives all possible values of x ?
The question asks you to find “all possible values of x,” which is just a fancy way of asking you to solve for x. First, make the 12 disappear from the left side of the inequality by subtracting 12 from both sides.
This leaves us with the inequality 4x > 4. Now get rid of the coefficient, which can be eliminated by dividing both sides of the inequality by 4.
4x > 4
Then reduce the fractions, noting that you can cancel out all the 4s in the equation.
x > 1
Therefore, the expression that provides all possible values of x is x > 1.
Refer back to Example 1
in Lesson 4.1 to see how
we handled single-
variable equations.
Take a look back at this page, where we solved the equation 4x + 12 = 16. Notice that the inequality you just solved is identical except for the inequality sign. Also notice that we solved the inequality in the same exact way as we solved the equation. The main difference with inequalities is that you find a range of values rather than a single value.
Another major difference between equations and inequalities is how you need to handle division or multiplication by a negative number. When working with inequalities, you must flip the orientation of the inequality sign if you multiply or divide by a negative number. Let’s see how this works in the following question.
EXAMPLE
If −8z + 10 > −4z − 6, what is the expression that gives all possible values of z ?
Just like in the previous example, start by moving the numbers and variables to opposite sides of the inequality. First, subtract 10 from both sides of the inequality.
Next, isolate z by adding 4z to both sides.
Now you’re left with −4z > −16. And this is where the difference between equations and inequalities comes into play. Whenever you divide or multiply by a negative number in an inequality, you must flip the orientation of the inequality sign. Here, you would divide both sides of the equation by −4, simplify, and flip the sign from greater than to less than.
−4z > −16
z < 4
Therefore, the expression that gives all possible values of z is z < 4.
Here, you need to find the value of x based on the given equation = 2. In order to isolate x and eliminate the fraction, multiply both sides of the equation by 2:2() = 2(2 ), so = 4. Next, eliminate the radical by squaring both sides of the equation ()^{2} = (4)^{2}, x = 16 × 2. Finally, simplify the equation to find that x = 16 × 2 and x = 32. Therefore, the correct answer is (D).
Are you multiplying or
dividing by a negative
number in an inequality?
Make sure you flip the
sign! Or, you can move
the variable to the side
of the equation where
it will be positive.
Not a fan of having to remember to flip the sign? If you want to avoid dividing or multiplying by a negative number in an inequality, simply move the variable to the side of the equation where it will be positive.
Consider the step from the previous question where we had the following: −4z > −16. Instead of dividing by −4, you could move −4z to the right side of the equation and −16 to the left side of the equation.
−4z > −16
16 > 4z
Now that all terms are positive, you can divide both sides of the equation by 4 to solve for z, without having to worry about flipping the sign.
16 > 4z
4 > z
Just like before, the expression that gives all possible values of z is 4 > z, or z < 4.
Writing Single-Variable Inequalities
Think back to when we discussed translating English to math. You use the same techniques to write single-variable inequalities as you do for single-variable linear equations. As a quick refresher, here are the words you should know for translating purposes.
English Word |
Math Equivalent |
is, are, were, did, does |
= |
of |
× |
out of |
÷ |
what |
variable, such as a, b, or c |
This table is the
same one from the
beginning of
Lesson 4.2. Keep these
English words and
their math equivalents
in mind when reading
math problems.
Now think back to our board game example from earlier. You’ve selected a game and are ready to play, but now you need to know how many people can play at once. The box says the game can have anywhere from 2 to 8 players, which means you can’t play by yourself, nor can you play with 15 of your classmates. So how would you represent this as an inequality? There are two parts to this that we need to address.
The first part of the problem is that you need at least 2 players. Using the letter p to represent the number of players, this can be expressed as p ≥ 2, indicating that you can have 2 or more players. The second part of the problem is that you cannot have more than 8 players. We can represent this as p ≤ 8.
Now combine this information to find that the number of players that can play the game at once is 2 ≤ p ≤ 8; you can have 2, 3, 4, 5, 6, 7, or 8 players, but you cannot have 1, 9, or 199 players.
Let’s try a couple of questions that deal with writing single-variable inequalities.
EXAMPLE
If six less than seven times a number is less than 15, what is the expression that gives all possible values of n ?
This question requires you to translate wordy English to math. You’re told that six less than seven times a number is less than 15. Using your translating skills, you can translate this phrase into the following:
(7 × n) − 6 < 15
Now, add 6 to both sides of the equation:
7n < 21
Finally, divide both sides of the equation by 7:
n < 3
Here is how you may see inequalities on the SAT.
If 7s − 14 < 4 + 6s, which of the following must be true?
A) s ≥ 17
B) s ≤ 18
C) s < 19
D) s < 18
Great work! Hopefully you’re noticing the similarities between translating equations and translating inequalities. The process is essentially the same, but instead of using an equals sign, you use an inequality sign. Let’s try another one.
EXAMPLE
In a card game involving cats, a white cat is worth two points, a black cat is worth three points, and a gray cat is worth five points. If a player collects more than five gray cats, a 20-point bonus is awarded. If Zed has two gray cat cards in his possession, and the number of points Zed has is represented by p, then, in terms of p, which of the following inequalities expresses the range of points Zed can receive after collecting two additional cats?
A) 20 ≤ p ≤ 40
B) 14 ≤ p ≤ 30
C) 14 ≤ p ≤ 20
D) 10 ≤ p ≤ 20
This question should look familiar to you, because we worked through a similar one on this page. (Remember Tashi?) Again, break the question down into smaller pieces to make it more manageable.
The question states that gray cats are worth five points each and Zed has two gray cats in his possession. Thus, Zed has 5 + 5 = 10 points so far. You are also told that Zed will collect two more cats. This is where it gets tricky. How do we know which type of cat Zed will collect? There are a few different scenarios:
White Cat + White Cat = 2 + 2 = 4
White Cat + Black Cat = 2 + 3 = 5
White Cat + Gray Cat = 2 + 5 = 7
Black Cat + Black Cat = 3 + 3 = 6
Black Cat + Gray Cat = 3 + 5 = 8
Gray Cat + Gray Cat = 5 + 5 = 10
However, when you are interested in a range of values, you really need to concern yourself with the least and greatest possible values. In this scenario, Zed will gain the least number of points, 4, by collecting two white cats and the greatest number of points, 10, by collecting two gray cats.
Now you can use the previous information—that Zed already has 10 points—to find the total range of point values. At a minimum, Zed can earn 10 + 4 = 14 points; p ≥ 14. At a maximum, Zed can earn 10 + 10 = 20 points; p ≤ 20. Thus, the inequality that expresses the range of points Zed can receive after four turns is 14 ≤ p ≤ 20, or (C).