Solve linear inequalities in one variable and represent solutions on a number line. | Linear inequalities in one or two variables | SAT Math

Solve one-variable inequalities and represent the solution set on a number line.

Core Idea

Solve a linear inequality exactly like an equation, but flip the inequality sign whenever you multiply or divide both sides by a negative number.

Understanding

Treat the inequality sign as an equals sign while you isolate the variable. Add, subtract, multiply, divide — all the same moves. The only twist: multiplying or dividing by a negative reverses the sign ( $>$ becomes $<$ , $\geq$ becomes $\leq$ ).

On a number line, an open circle means the endpoint is not included (strict $<$ or $>$ ), and a closed circle means it is included ( $\leq$ or $\geq$ ). Shade the direction that contains all valid values.

For example, solving $- 3 𝑥 + 6 \leq 15$ gives $- 3 𝑥 \leq 9$ , and dividing by $- 3$ flips the sign to $𝑥 \geq - 3$ . On the number line: closed circle at $- 3$ , shade to the right.

Step by Step

Simplify each side (distribute, combine like terms).
Use addition or subtraction to move all variable terms to one side and constants to the other.
Divide or multiply to isolate the variable — if by a negative number, flip the inequality sign.
Represent the solution on a number line: open circle for strict inequalities, closed circle for inclusive ones, shade the valid direction.

Misconceptions

Forgetting to flip the inequality sign when multiplying or dividing by a negative — this is the single most common error.
Using a closed circle on the number line for a strict inequality (< or >), or an open circle for ≤ or ≥.
Flipping the sign when subtracting a negative number — the flip rule only applies to multiplication and division.

Question

Worked Example

What is the solution to $4 - 2 𝑥 > 10$ ?

Select an answer to see the explanation