How to Solve a Linear Inequality

Linear inequalities are almost identical to linear equations. Instead of the answer being a number, the answer is an interval. When you solve an inequality, you must find what $x$ must be greater than, greater than or equal to, less than, or less than or equal to.

Rule

Inequality Signs

$>$ means “greater than”

$\geq$ means “greater than or equal to”

$<$ means “less than”

$\leq$ means “less than or equal to”

Example 1

Solve the inequality $x + 3 < 8$

\begin{array}{l} x + 3 & < 8 \\ x & < 8 - 3 \\ x & < 5 \end{array}

The answer tells you that

x

must be less than 5 for the inequality to be true.

There is one important difference between solving linear inequalities and linear equations: When you multiply or divide by a negative number, you have to turn the inequality! Why? Take a look at the following example:

Example 2

You might agree that

5 > - 10

because $- 10$ is further left on the real line than 5. If you multiply both sides with $- 1$ , you get:

\begin{array}{l} 5 & > - 10 \\ (- 1) \cdot 5 & > - 10 \cdot (- 1) \\ - 5 & > 10 \end{array}

Now it says that $- 5$ is greater than 10. That doesn’t make sense! 10 is further to the right on the number line than $- 5$ , and is therefore a greater number.

Due to this issue with the sign, the inequality sign must be turned for the expression to still be true.

Rule

Multiply or Divide by Negative Numbers

When you multiply or divide by a negative number, you have to TURN the inequality.

Example 3

Solve the inequality $- x - 12 < 14$

\begin{array}{l} - x - 12 & < 14 \\ - x & < 14 + 12 \\ (- 1) \cdot - x & > 26 \cdot (- 1) \\ x & > - 26 \end{array}

The answer tells you that

x

must be greater than

- 26

for the inequality to be true.

Example 4

Solve the inequality $2 (x + 2) - 3 x > - (- x - 3) + 4$

\begin{array}{l} 2 (x + 2) - 3 x & > - (- x - 3) + 4 \\ 2 x + 4 - 3 x & > x + 3 + 4 \\ 2 x - 3 x - x & > 3 + 4 - 4 \\ - 2 x & > 3 \\ \frac{-2 x}{-2} & < \frac{3}{- 2} \\ x & < - \frac{3}{2} \end{array}

The answer tells you that

x

must be less than

- \frac{3}{2}

for the inequality to be true.

Example 5

Solve the inequality $2 x - 3 (2 x - 5) \geq 2 (3 x - 2) + 5 x$

\begin{matrix} 2 x - 3 (2 x - 5) \geq 2 (3 x - 2) + 5 x \\ 2 x - 6 x + 15 \geq 6 x - 4 + 5 x \\ 2 x - 6 x - 6 x - 5 x \geq - 4 - 15 \\ - 15 x \geq - 19 \\ \frac{-15 x}{-15} \geq \frac{- 19}{- 15} \\ x \leq \frac{19}{15} \end{matrix}

\begin{array}{l} 2 x - 3 (2 x - 5) & \geq 2 (3 x - 2) + 5 x \\ 2 x - 6 x + 15 & \geq 6 x - 4 + 5 x \\ 2 x - 6 x - 6 x - 5 x & \geq - 4 - 15 \\ - 15 x & \geq - 19 \\ \frac{-15 x}{-15} & \geq \frac{- 19}{- 15} \\ x & \leq \frac{19}{15} \end{array}

The answer tells you that

x

must be less than or equal to

\frac{19}{15}

for the inequality to be true.

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