Math Core

You already know how to solve $x + 5 = 2$ in one step. Solving $x + 5 > 2$ uses the same move, and the answer is a whole range of numbers instead of one. There's just one place where inequalities behave differently from equations, and this lesson shows you exactly where.

Adding and subtracting

Think of an inequality as a scale that is tipped: one side is heavier. If you add the same weight to both sides, or take the same weight off both sides, the heavier side stays heavier. So you can add or subtract any number on both sides, and the symbol stays the same.

Worked example: Undoing addition and subtraction

Solve each inequality and graph the solutions.

$x + 5 > 2$
$x - 4 \le -6$

1. Subtract $5$ from both sides: $x > 2 - 5$ , so $x > -3$ .

x > -3

2. Add $4$ to both sides: $x \le -6 + 4$ , so $x \le -2$ .

x ≤ -2

Check part 1: try $0$ , which is shaded: $0 + 5 = 5$ , and $5 > 2$ . ✓ Try $-4$ , which is not: $-4 + 5 = 1$ , and $1 > 2$ is false. ✓

Multiplying and dividing by a positive number

Multiplying or dividing both sides by a positive number also keeps the inequality true. If $6 < 10$ , then $6 \div 2 < 10 \div 2$ , since $3 < 5$ . The numbers shrink, but they stay in the same order.

Worked example: Dividing by a positive

Solve $3x \ge -12$ .

Divide both sides by $3$ . Since $3$ is positive, keep the symbol:

\begin{aligned} 3x &\ge -12 \\ x &\ge -4 \end{aligned}

Notice that the right side is negative. That doesn't matter. Only the number you divide by decides whether to flip, and $3$ is positive.

x ≥ -4

Multiplying and dividing by a negative number

Here is the one surprise. Start with a true statement, $3 < 5$ , and multiply both sides by $-1$ . You get $-3$ and $-5$ . But $-3$ is to the right of $-5$ on the number line, so now $-3 > -5$ .

Multiplying by a negative number reflects every number across $0$ , like a mirror. Numbers that were in order from left to right end up in the reverse order. To keep the statement true, the symbol has to turn around too.

The flip rule

When you multiply or divide both sides of an inequality by a negative number, reverse the symbol: $<$ and $>$ swap, and $\le$ and $\ge$ swap.

Adding or subtracting (even a negative number) never flips the symbol. Multiplying or dividing by a positive number never flips it either.

Worked example: Dividing by a negative

Solve $-2x < 10$ .

Divide both sides by $-2$ and flip $<$ to $>$ :

\begin{aligned} -2x &< 10 \\ \dfrac{-2x}{-2} &> \dfrac{10}{-2} \\ x &> -5 \end{aligned}

x > -5

Check: try $0$ : $-2(0) = 0$ , and $0 < 10$ . ✓ Try $-6$ , which is not shaded: $-2(-6) = 12$ , and $12 < 10$ is false. ✓

Worked example: Multiplying by a negative

Solve $\dfrac{x}{-3} \ge 2$ .

To undo dividing by $-3$ , multiply both sides by $-3$ and flip the symbol:

\begin{aligned} \dfrac{x}{-3} &\ge 2 \\ x &\le -6 \end{aligned}

Check: the boundary $-6$ gives $\dfrac{-6}{-3} = 2$ , and $2 \ge 2$ is true, so the closed dot is right. Try $-9$ : $\dfrac{-9}{-3} = 3$ , and $3 \ge 2$ . ✓ Try $0$ : $0 \ge 2$ is false, and $0$ is not shaded. ✓

x ≤ -6

Common mistake

The flip depends on the number you multiply or divide by, not on where the negative signs appear. In $3x \ge -12$ you divide by positive $3$ , so no flip, even though $-12$ is negative. In $-2x < 10$ you divide by $-2$ , so you do flip. Ask yourself: "Am I multiplying or dividing by a negative?" Only a yes means flip.

A word problem

Worked example: Packing a suitcase

An airline allows a checked bag to weigh at most 50 pounds. Jordan's bag already weighs 38 pounds. How much more weight can Jordan add?

Let $w$ be the extra weight in pounds. The total can be at most 50:

\begin{aligned} 38 + w &\le 50 \\ w &\le 12 \end{aligned}

Jordan can add up to 12 pounds. Weight can't be negative, so in this story the answer really means any weight from $0$ to $12$ pounds.

Tip

Always check with a number from your answer. Plug it into the original inequality. If it makes the original false, you probably forgot to flip (or flipped when you shouldn't have).