Math Core

You already know how to solve equations by undoing operations until the variable stands alone. Inequalities are solved almost the same way, with one important twist. Once you know when that twist happens, you can solve any one- or two-step inequality and graph its solutions.

What stays the same

An inequality is like a scale that is tipped to one side. If you add the same weight to both pans, or take the same weight away, the scale stays tipped the same way. Numbers behave like that too.

Start with a true statement, $3 < 8$ :

Add $5$ to both sides: $8 < 13$ . Still true.
Subtract $10$ from both sides: $-7 < -2$ . Still true.
Multiply both sides by $2$ : $6 < 16$ . Still true.
Divide both sides by $4$ : $0.75 < 2$ . Still true.

So adding, subtracting, and multiplying or dividing by a positive number all keep the inequality symbol exactly as it is.

What changes: multiplying by a negative

Now multiply both sides of $3 < 8$ by $-1$ . You get $-3$ and $-8$ . But $-3$ is to the right of $-8$ on the number line, so $-3 > -8$ . The original symbol $<$ would give a false statement; the true one is $>$ .

Multiplying by −1 reflects each point across 0, so the order of the two numbers reverses.

Multiplying by a negative number reflects every number across $0$ . The number that used to be farther right ends up farther left, so the order reverses. Dividing by a negative does the same thing, since dividing by $-4$ is the same as multiplying by $-\dfrac{1}{4}$ .

Properties of inequality

You may add or subtract the same number on both sides. The symbol stays the same.
You may multiply or divide both sides by the same positive number. The symbol stays the same.
If you multiply or divide both sides by a negative number, you must reverse the symbol: $<$ becomes $>$ , $\le$ becomes $\ge$ , and so on.

One-step inequalities

Worked example: Adding, subtracting, multiplying, dividing

Solve each inequality and graph the solutions.

$x + 7 > 12$
$\dfrac{m}{4} \le -2$
$-5y \ge 30$

Solutions.

Subtract $7$ from both sides: $x > 5$ . Open circle at $5$ , shade right.

Multiply both sides by $4$ , a positive number, so the symbol stays: $m \le -8$ . Closed circle at $-8$ , shade left.

Divide both sides by $-5$ . The divisor is negative, so reverse the symbol:

\frac{-5y}{-5} \le \frac{30}{-5} \quad\Longrightarrow\quad y \le -6.

Check a value from the answer: $y = -7$ gives $-5(-7) = 35$ , and $35 \ge 30$ is true. Check a value outside it: $y = 0$ gives $0 \ge 30$ , false. The flipped answer is right.

Common mistake

Reverse the symbol only when you multiply or divide by a negative number. A negative number somewhere in the problem is not a reason to flip. In $3x < -12$ you divide by positive $3$ , so the answer is $x < -4$ with no flip. In $x - 6 > -2$ you add $6$ , so the answer is $x > 4$ , again with no flip.

Two-step inequalities

Two-step inequalities are solved in the same order as two-step equations: undo addition or subtraction first, then undo multiplication or division.

Worked example: A two-step inequality

Solve $3x - 5 \ge 16$ .

\begin{aligned} 3x - 5 &\ge 16 \\ 3x &\ge 21 && \text{add } 5 \\ x &\ge 7 && \text{divide by } 3 \text{ (positive, no flip)} \end{aligned}

The solutions are all numbers $7$ or greater.

Worked example: A negative coefficient

Solve $9 - 2x > 15$ and graph the solutions.

\begin{aligned} 9 - 2x &> 15 \\ -2x &> 6 && \text{subtract } 9 \\ x &< -3 && \text{divide by } -2 \text{ and reverse} \end{aligned}

x < -3

Check. The boundary $x = -3$ should make the two sides equal: $9 - 2(-3) = 15$ . Good. A number in the shaded part, $x = -4$ , should make it true: $9 - 2(-4) = 17 > 15$ . Good.

Tip

Check every answer with two numbers. The boundary should make both sides equal, which confirms the number. A test point on the shaded side should make the original inequality true, which confirms the direction. A forgotten flip is caught instantly by the test point.

Inequalities in word problems

Word problems lead to inequalities whenever a limit is involved: a budget, a capacity, a minimum score.

Worked example: Staying under budget

A ride-share costs $4 to start plus $1.50 per mile. You have $25. How many miles can you ride?

Let $d$ be the number of miles. The total cost must be at most $25:

\begin{aligned} 4 + 1.5d &\le 25 \\ 1.5d &\le 21 && \text{subtract } 4 \\ d &\le 14 && \text{divide by } 1.5 \end{aligned}

You can ride at most $14$ miles. Check: $4 + 1.5(14) = 4 + 21 = 25$ , exactly your budget.

Notice the steps are the same as for an equation. The inequality just keeps track of the fact that anything less than $14$ miles is fine too.