Use the addition and multiplication properties to solve algebraic inequalities and express their solutions graphically and with interval notation
Solve inequalities with addition and subtraction
You can solve most inequalities using inverse operations as you did for solving equations. This is because when you add or subtract the same value from both sides of an inequality, you have maintained the inequality. These properties are outlined in the box below.
Addition and Subtraction Properties of Inequality
If [latex]a>b[/latex],then [latex]a+c>b+c[/latex].
If [latex]a>b[/latex], then [latex]a−c>b−c[/latex].
Because inequalities have multiple possible solutions, representing the solutions graphically provides a helpful visual of the situation, as we saw in the last section. The example below shows the steps to solve and graph an inequality and express the solution using interval notation.
Example
Solve for x.
[latex] {x}+3\lt{5}[/latex]
Show Solution
It is helpful to think of this inequality as asking you to find all the values for x, including negative numbers, such that when you add three you will get a number less than 5.
Isolate the variable by subtracting 3 from both sides of the inequality.
Answer
Inequality: [latex]x<2[/latex]
Interval: [latex]\left(-\infty, 2\right)[/latex]
Graph:
The line represents all the numbers to which you can add 3 and get a number that is less than 5. There’s a lot of numbers that solve this inequality!
Just as you can check the solution to an equation, you can check a solution to an inequality. First, you check the end point by substituting it in the related equation. Then you check to see if the inequality is correct by substituting any other solution to see if it is one of the solutions. Because there are multiple solutions, it is a good practice to check more than one of the possible solutions. This can also help you check that your graph is correct.
The example below shows how you could check that [latex]x<2[/latex]is the solution to [latex]x+3<5[/latex].
Example
Check that [latex]x<2[/latex]is the solution to [latex]x+3<5[/latex].
Show Solution
Substitute the end point 2 into the related equation, [latex]x+3=5[/latex].
[latex]x<2[/latex] is the solution to [latex]x+3<5[/latex].
The following examples show inequality problems that include operations with negative numbers. The graph of the solution to the inequality is also shown. Remember to check the solution. This is a good habit to build!
Example
Solve for x: [latex]x-10\leq-12[/latex]
Show Solution
Isolate the variable by adding 10 to both sides of the inequality.
Inequality: [latex]x\leq-2[/latex]
Interval: [latex]\left(-\infty,-2\right][/latex]
Graph: Notice that a closed circle is used because the inequality is “less than or equal to” [latex]\left(\leq\right)[/latex]. The blue arrow is drawn to the left of the point [latex]−2[/latex] because these are the values that are less than [latex]−2[/latex].
Check the solution to [latex]x-10\leq -12[/latex]
Show Solution
Substitute the end point [latex]−2[/latex] into the related equation [latex]x-10=−12[/latex]
[latex] \displaystyle a\,>\,0[/latex] is the solution to [latex]a-17>-17[/latex]
The previous examples showed you how to solve a one-step inequality with the variable on the left hand side. The following video provides examples of how to solve the same type of inequality.
What would you do if the variable were on the right side of the inequality? In the following example, you will see how to handle this scenario.
Example
Solve for x: [latex]4\geq{x}+5[/latex]
Show Solution
Isolate the variable by adding 10 to both sides of the inequality.
Rewrite the inequality with the variable on the left – this makes writing the interval and drawing the graph easier.
[latex]x\le{-1}[/latex]
Note how the the pointy part of the inequality is still directed at the variable, so instead of reading as negative one is greater or equal to x, it now reads as x is less than or equal to negative one.
Answer
Inequality: [latex]x\le{-1}[/latex] This can also be written as
Interval: [latex]\left(-\infty,-1\right][/latex]
Graph: Notice that a closed circle is used because the inequality is “less than or equal to” . The blue arrow is drawn to the left of the point [latex]−1[/latex] because these are the values that are less than [latex]−1[/latex].
Check the solution to [latex]4\geq{x}+5[/latex]
Show Solution
Substitute the end point [latex]−1[/latex] into the related equation [latex]4=x+5[/latex]
[latex]x\le{-1}[/latex] is the solution to [latex]4\geq{x}+5[/latex]
The following video show examples of solving inequalities with the variable on the right side.
Solve inequalities with multiplication and division
Solving an inequality with a variable that has a coefficient other than 1 usually involves multiplication or division. The steps are like solving one-step equations involving multiplication or division EXCEPT for the inequality sign. Let’s look at what happens to the inequality when you multiply or divide each side by the same number.
Let’s start with the true statement:
[latex]10>5[/latex]
Let’s try again by starting with the same true statement:
[latex]10>5[/latex]
Next, multiply both sides by the same positive number:
[latex]10\cdot 2>5\cdot 2[/latex]
This time, multiply both sides by the same negative number:
20 is greater than 10, so you still have a true inequality:
[latex]20>10[/latex]
Wait a minute! [latex]−20[/latex] is not greater than [latex]−10[/latex], so you have an untrue statement.
[latex]−20>−10[/latex]
When you multiply by a positive number, leave the inequality sign as it is!
You must “reverse” the inequality sign to make the statement true:
[latex]−20<−10[/latex]
Caution! When you multiply or divide by a negative number, “reverse” the inequality sign. Whenever you multiply or divide both sides of an inequality by a negative number, the inequality sign must be reversed in order to keep a true statement. These rules are summarized in the box below.
Multiplication and Division Properties of Inequality
Keep in mind that you only change the sign when you are multiplying and dividing by a negative number. If you add or subtract by a negative number, the inequality stays the same.
There was no need to make any changes to the inequality sign because both sides of the inequality were divided by positive 3. In the next example, there is division by a negative number, so there is an additional step in the solution!
Example
Solve for x. [latex]−2x>6[/latex]
Show Solution
Divide each side of the inequality by [latex]−2[/latex] to isolate the variable, and change the direction of the inequality sign because of the division by a negative number.
Graph:
Because both sides of the inequality were divided by a negative number, [latex]−2[/latex], the inequality symbol was switched from > to <.
The following video shows examples of solving one step inequalities using the multiplication property of equality where the variable is on the left hand side.
Think About It
Before you read the solution to the next example, think about what properties of inequalities you may need to use to solve the inequality. What is different about this example from the previous one? Write your ideas in the box below.
Solve for x. [latex]-\frac{1}{2}>-12x[/latex]
Show Solution
This inequality has the variable on the right hand side, which is different from the previous examples. Start the solution process as before, and at the end, you can move the variable to the left to write the final solution.
Divide both sides by [latex]-12[/latex] to isolate the variable. Since you are dividing by a negative number, you need to change the direction of the inequality sign.
Inequality: [latex]\frac{1}{24}\lt{x}[/latex] This can also be written with the variable on the left as [latex]x\gt\frac{1}{24}[/latex]. Writing the inequality with the variable on the left requires a little thinking, but helps you write the interval and draw the graph correctly.
The following video gives examples of how to solve an inequality with the multiplication property of equality where the variable is on the right hand side.
Summary
Solving inequalities is very similar to solving equations, except you have to reverse the inequality symbols when you multiply or divide both sides of an inequality by a negative number. There are three ways to represent solutions to inequalities: an interval, a graph, and an inequality. Because there is usually more than one solution to an inequality, when you check your answer you should check the end point and one other value to check the direction of the inequality.
Licenses and Attributions
CC licensed content, Original
Revision and Adaptation. Provided by: Lumen Learning. License: CC BY: Attribution
CC licensed content, Shared previously
Unit 10: Solving Equations and Inequalities, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology and Education. Located at: http://nrocnetwork.org/dm-opentext. License: CC BY: Attribution
Ex: Solving One Step Inequalities by Adding and Subtracting (Variable Left Side). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. Located at: https://youtu.be/1Z22Xh66VFM. License: CC BY: Attribution
Ex: Solving One Step Inequalities by Adding and Subtracting (Variable Right Side). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. Located at: https://youtu.be/RBonYKvTCLU. License: CC BY: Attribution
Ex: Solve One Step Linear Inequality by Dividing (Variable Left). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. Located at: https://youtu.be/IajiD3R7U-0. License: CC BY: Attribution
Ex: Solve One Step Linear Inequality by Dividing (Variable Right). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. Located at: https://youtu.be/s9fJOnVTHhs. License: CC BY: Attribution
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Caution! When you multiply or divide by a negative number, “reverse” the inequality sign. Whenever you multiply or divide both sides of an inequality by a negative number, the inequality sign must be reversed in order to keep a true statement. These rules are summarized in the box below.
