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Solving Linear Inequalities

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This lesson is part of a series of lessons for the quantitative reasoning section of the GRE revised General Test. In this lesson, we will learn how to solve linear inequalities.

Linear Inequalities

A mathematical statement that uses one of the following inequality signs is called an inequality.

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

Inequalities can involve variables and are similar to equations, except that the two sides are related by one of the inequality signs instead of the equality sign used in equations.

For example, the inequality x + 2 ≤ 7 is a linear inequality in one variable, which states that “x + 2 is less than or equal to 7.”

To solve an inequality means to find the set of all values of the variable that make the inequality true. This set of values is also known as the solution set of an inequality. Two inequalities that have the same solution set are called equivalent inequalities.

The procedure used to solve a linear inequality is similar to that used to solve a linear equation, which is to simplify the inequality by isolating the variable on one side of the inequality, using the following two rules.
• When the same constant is added to or subtracted from both sides of an inequality, the direction of the inequality is preserved and the new inequality is equivalent to the original.
• When both sides of the inequality are multiplied or divided by the same nonzero constant, the direction of the inequality is preserved if the constant is positive but the direction is reversed if the constant is negative. In either case, the new inequality is equivalent to the original.

For example,

Solve the inequality 7 – x < 9

7 – x < 9
7 – x – 7 < 9 – 7
x < 2
x > –2 (remember to reverse the symbol when multiplying by –1 )

For example,


The following videos show some examples of how to solve linear inequalities.


Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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