Linear Inequalities

Quick Reference

Operation	Effect on Inequality Direction
Add or subtract any quantity	Unchanged
Multiply or divide by a positive number	Unchanged
Multiply or divide by a negative number	Reverses (e.g. becomes )
Multiply or divide by an expression of unknown sign	Never do this

To solve inequalities, we rely on fundamental properties discussed in Section on Inequalities. Two of the most important are:

• We can add (or subtract) the same quantity from both sides without changing the direction of the inequality.
• We can multiply (or divide) both sides by a positive quantity, without changing the direction of the inequality.
• If we multiply or divide both sides of an inequality by a negative quantity, the direction of the inequality reverses (see Section on Inequalities).

Linear inequalities are often easy to solve. We just need to isolate the variable on one side of the inequality sign.

• For the last example, note thatmeansIn fact, we have to solve two inequalities.

Frequently Asked Questions

What is a linear inequality?

A linear inequality is a statement that compares two linear expressions using one of the symbols , $\le$, , or $\ge$. For example, $7x-5\ge 4x+4$ is a linear inequality. Its solution is typically a range of values rather than a single number.

Why does the inequality reverse when you divide by a negative number?

Because multiplying or dividing by a negative number flips the relative order of numbers on the number line. For example, , but . If you divide the inequality $-5x\le -5$ by $-5$ without reversing, you would incorrectly conclude $x\le 1$; the correct answer is $x\ge 1$.

What is the difference between a strict inequality (, ) and a non-strict inequality ($\le$, $\ge$)?

A strict inequality like excludes the endpoint (3 is not a solution). A non-strict inequality like $x\le 3$ includes the endpoint (3 is a solution). In interval notation, strict inequalities use parentheses, while non-strict inequalities use brackets: $(-\mathrm{\infty },3)$ vs. $(-\mathrm{\infty },3]$.

How do you solve a compound (double) inequality like ?

Apply all algebraic operations simultaneously to all three parts of the inequality. In the example, multiplying all three parts by 2 gives . Subtracting 7 from all three parts gives . Dividing all three parts by $-5$ (and reversing both inequality signs) gives , or equivalently .

How do I express the solution of a linear inequality in interval notation?

Write the solution as an interval. Use a square bracket $[$ or $]$ if the endpoint is included ($\le$ or $\ge$), and a parenthesis $($ or $)$ if it is excluded ( or ). For example, $x\ge 3$ becomes $[3,+\mathrm{\infty })$, and becomes $[1, 3)$.