Math Core

A school club has $40 to spend on T-shirts. The club doesn't need to spend exactly $40. It just can't spend more. Questions like "how many can we afford?" lead to inequalities, and you can solve them almost exactly like two-step equations. There's just one new rule to learn.

Solving works (almost) like equations

You can add or subtract the same number on both sides of an inequality, and it stays true. You can also multiply or divide both sides by the same positive number.

Worked example: A familiar-looking inequality

Solve $3x + 4 > 19$ and graph the solutions.

Use the same steps as for $3x + 4 = 19$ : subtract $4$ , then divide by $3$ .

\begin{aligned} 3x + 4 &> 19 \\ 3x &> 15 \\ x &> 5 \end{aligned}

Every number greater than $5$ is a solution. The open circle shows that $5$ itself is not included.

x > 5

Check: try a number in the shaded part, like $6$ : $3(6) + 4 = 22$ , and $22 > 19$ . ✓ Try one outside, like $2$ : $3(2) + 4 = 10$ , and $10 > 19$ is false. ✓

The one new rule: negatives flip the sign

Start with a true statement: $2 < 5$ . Now multiply both sides by $-1$ . You get $-2$ and $-5$ . On a number line, $-2$ is to the right of $-5$ , so

-2 > -5.

The numbers switched places when they became negative, so the inequality symbol had to turn around to stay true.

Flip the symbol for negatives

When you multiply or divide both sides of an inequality by a negative number, reverse the inequality symbol.

$<$ becomes $>$ , $>$ becomes $<$ , $\le$ becomes $\ge$ , and $\ge$ becomes $\le$ .

Adding or subtracting a negative number does not flip the symbol.

Worked example: Dividing by a negative

Solve $-4x + 7 \le 23$ and graph the solutions.

\begin{aligned} -4x + 7 &\le 23 \\ -4x &\le 16 \\ \dfrac{-4x}{-4} &\ge \dfrac{16}{-4} \qquad \text{(divide by } -4 \text{, flip the symbol)} \\ x &\ge -4 \end{aligned}

The closed dot shows that $-4$ is included.

x ≥ -4

Check: try $0$ : $-4(0) + 7 = 7$ , and $7 \le 23$ . ✓ Try $-5$ , which is outside the shading: $-4(-5) + 7 = 27$ , and $27 \le 23$ is false. ✓

Worked example: A negative fraction

Solve $5 - \dfrac{x}{2} \ge 1$ .

Subtract $5$ from both sides. The $x$ term is $-\dfrac{x}{2}$ , so multiply both sides by $-2$ and flip the symbol.

\begin{aligned} 5 - \dfrac{x}{2} &\ge 1 \\ -\dfrac{x}{2} &\ge -4 \\ x &\le 8 \end{aligned}

x ≤ 8

Check: try $4$ : $5 - 2 = 3$ , and $3 \ge 1$ . ✓

Common mistake

Forgetting to flip is the most common mistake. In $-4x \le 16$ , dividing by $-4$ without flipping gives $x \le -4$ . Test $x = -5$ : $-4(-5) = 20$ , and $20 \le 16$ is false. So $x \le -4$ can't be right. Always test one number from your answer in the original inequality.

Word problems with inequalities

Watch for phrases like "at most," "no more than," "at least" and "a minimum of." After solving, think about what the solutions mean in the story.

Worked example: Buying T-shirts

The club has $40. T-shirts cost $12 each, plus a single $4 shipping charge. How many shirts can the club buy?

Let $s$ be the number of shirts. The total cost can be at most $40$ dollars:

\begin{aligned} 12s + 4 &\le 40 \\ 12s &\le 36 \\ s &\le 3 \end{aligned}

The math says $s \le 3$ . But you can't buy $2.5$ shirts or $-1$ shirt. So the club can buy $0$ , $1$ , $2$ or $3$ shirts, and 3 shirts is the most. Check: $3$ shirts cost $12(3) + 4 = 40$ dollars, which is allowed. $4$ shirts would cost $52$ dollars, which is too much.

Tip

To decide between an open circle and a closed dot, look at the symbol. $<$ and $>$ leave the endpoint out (open circle). $\le$ and $\ge$ include it (closed dot).