I - Solve Inequalities by Adding or Subtracting Lesson

Solve Inequalities by Adding or Subtracting

In this lesson, you will use your equation skills as a basis for learning how to solve inequalities. You will see that the process for addition and subtraction of inequalities is the same as for equations as you continue to use inverse operations to solve them. There is one big difference in the outcome of an equation and an inequality - In an equation, you find the solution and in an inequality, you will find a solution set.

When you solve inequalities, you will use one of the following symbols:

Symbol	Represents
<	less than
>	greater than
≤	less than or equal to
≥	greater than or equal to

To solve an inequality, you follow the same rules as when solving an equation. The solution of an inequality is a set of numbers known as the solution set. When you solve an equation, you check your work by substituting the number for the variable and solving the problem. When you solve an inequality, you select any number in the solution set and substitute that number for the variable.

minus text annotation indicator

Example 1

What is the solution set for this inequality?

$LaTeX: x-8<11\\ \text{Add 8 to both sides.}\\ x-8< 11\\ +8\:\:+8\\ x< 19$ $x-8<11\\ \text{Add 8 to both sides.}\\ x-8< 11\\ +8\:\:+8\\ x< 19$

x can be ANY number less than 19, so the solution set is infinite.

Check: Select a number less than 19 and substitute for x.

10 < 19 Correct!

plus text annotation indicator

Example 2

$LaTeX: x+8>11\\ \text{Subtract 8 from both sides.}\\ x+8> 11\\ -8\:\:-8\\ x> 3$ $x+8>11\\ \text{Subtract 8 from both sides.}\\ x+8> 11\\ -8\:\:-8\\ x> 3$

x can be ANY number greater than 3, so the solution set is infinite.

Check: Select a number greater than 3 and substitute for x.

5 > 3 Correct!

check this out!

Example 3

When you use these symbols: (≤, ≥)your solution set contains the number used to solve the inequality. If I replaced the symbol in example 2 with this symbol, 3 would be part of the solution set.

$LaTeX: x+8\ge 11\\ \text{Add 8 to both sides.}\\ x+8\ge 11\\ -8\:\:-8\\ x\ge 3$ $x+8\ge 11\\ \text{Add 8 to both sides.}\\ x+8\ge 11\\ -8\:\:-8\\ x\ge 3$

x can be ANY number greater than or equal to 3, so the solution set is infinite.

Check: Select a number 3 or greater and substitute for x.

18 ≥ 3 Correct!

Now, take a few minutes to watch this video demonstrating the concept before you move forward.

How do you write and solve an inequality from a word problem?

Investigate

person on ice skates Monday's high temperature was 20 °F. Tuesday's high temperature was forecast to be no more than 8 °F warmer than Monday's high temperature. According to the forecast, what are the possible high temperatures for Tuesday?

Strategy

Now is the time to translate the words to math, using key words.

The words "no more than" indicate ≤ 8.

The word "was" indicates where to place the inequality sign.

Since Tuesday's temps could be higher, subtract 20 from Tuesday's temp.(t)

t - 20 ≤ 8

Solution

The inequality t - 20 ≤ 8 represents the possible high temperatures for Tuesday. Once you have written the inequality, use inverse operations and solve, just as you would solve a simple equation

$LaTeX: t-20\le8 \\ \text{Add 20 to both sides of the inequality.}\\ \:\:+20\:\:+20\\ t\le 28$ $t-20\le8 \\ \text{Add 20 to both sides of the inequality.}\\ \:\:+20\:\:+20\\ t\le 28$

This means the possible high temperatures for Tuesday are 28°F or less.

Investigate

image of bus

Let's try another type of word problem!

The Y Club is sponsoring a trip to the Fox Theatre for a performance, however there must be at least 130 members who will sign up or they cannot pay for the buses. So far, 98 members have signed up for the trip. How many more members must sign up to make this trip possible?

Strategy

First, translate the words to math, using key words and phrases.

The words "at least 130" indicate ≥ 130.

The words "how many more" indicate addition.

You will need to combine (or add) the total members, m

m + 98 ≥ 130

Solution

$LaTeX: m+98\ge 130\\ \text{Subtract 98 from both sides of the inequality.}\\ m+98\ge 130\\ -98\:\:-98\\ m\ge 32$ $m+98\ge 130\\ \text{Subtract 98 from both sides of the inequality.}\\ m+98\ge 130\\ -98\:\:-98\\ m\ge 32$

This means that at least 32 Y Club members must sign up in order for the trip to be possible.

Solve Addition and Subtraction Inequalities Practice

Solve Addition and Subtraction Inequalities Homework

Now that you have spent some time learning strategies for solving inequalities using addition or subtraction, you are ready to complete your Inequalities: Addition and Subtraction Homework. Download your homework by CLICKING HERE Links to an external site..

Once you have completed your homework, AND MAKE SURE YOU ATTEMPTED AND WORKED THE PROBLEMS OUT ON YOUR OWN, click here to download your homework key Links to an external site..

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