Math Core

Most real limits involve a starting amount plus a rate: a phone plan with a monthly fee plus a charge per gigabyte, or savings that grow by the same amount each week. Those situations give two-step inequalities. You solve them with the same plan you use for two-step equations, plus the flip rule from the last lesson.

The plan

Undo the operations in reverse order, just like with an equation:

Add or subtract to get the variable term alone on one side.
Multiply or divide to get the variable alone. If that number is negative, flip the symbol.
Check a number from your answer in the original inequality, then graph.

Two steps, one possible flip

Solve a two-step inequality exactly like a two-step equation. The only difference comes in step 2: if you multiply or divide by a negative number, reverse the inequality symbol.

Worked example: No flip needed

Solve $5x - 3 \ge 17$ and graph the solutions.

\begin{aligned} 5x - 3 &\ge 17 \\ 5x &\ge 20 \qquad \text{(add } 3\text{)} \\ x &\ge 4 \qquad \text{(divide by } 5\text{)} \end{aligned}

x ≥ 4

Check: try $6$ : $5(6) - 3 = 27$ , and $27 \ge 17$ . ✓ Try $2$ , which is not shaded: $5(2) - 3 = 7$ , and $7 \ge 17$ is false. ✓

Worked example: The variable term is negative

Solve $10 - 3x > 1$ .

The $x$ term is $-3x$ . Subtract $10$ first, then divide by $-3$ and flip.

\begin{aligned} 10 - 3x &> 1 \\ -3x &> -9 \qquad \text{(subtract } 10\text{)} \\ x &< 3 \qquad \text{(divide by } -3\text{, flip)} \end{aligned}

x < 3

Check: try $0$ : $10 - 0 = 10$ , and $10 > 1$ . ✓ Try $4$ : $10 - 12 = -2$ , and $-2 > 1$ is false. ✓

Common mistake

Watch out for $10 - 3x$ . The number in front of $x$ is $-3$ , not $3$ , because the minus sign belongs to the $3x$ term. If you divide by $3$ instead, you won't flip, and you'll get $x > 3$ . Test $x = 4$ in the original: $10 - 12 = -2$ , and $-2 > 1$ is false. So $x > 3$ is wrong.

Worked example: A fraction

Solve $\dfrac{x}{4} + 2 \le -1$ .

\begin{aligned} \dfrac{x}{4} + 2 &\le -1 \\ \dfrac{x}{4} &\le -3 \qquad \text{(subtract } 2\text{)} \\ x &\le -12 \qquad \text{(multiply by } 4\text{)} \end{aligned}

You multiplied by positive $4$ , so there's no flip, even though the answer is negative.

Check: try $-16$ : $\dfrac{-16}{4} + 2 = -4 + 2 = -2$ , and $-2 \le -1$ . ✓

Parentheses: a preview of Algebra 1

Sometimes the inequality has parentheses, like $2(x + 3) > 14$ . You can divide both sides by $2$ first (it's positive, so no flip), or use the distributive property. Both work:

\begin{aligned} 2(x + 3) &> 14 \\ x + 3 &> 7 \qquad \text{(divide by } 2\text{)} \\ x &> 4 \end{aligned}

Or: $2x + 6 > 14$ , so $2x > 8$ and $x > 4$ . Same answer. In Algebra 1 you'll solve longer inequalities this way, with the variable on both sides.

Word problems

Translate the words into an inequality, solve it, and then ask what the answer means in the story. Often only whole numbers make sense.

Worked example: A phone plan

A phone plan costs $25 per month plus $4 for each gigabyte of data. Priya wants her bill to be no more than $45. How many gigabytes can she use?

Let $g$ be the number of gigabytes.

\begin{aligned} 25 + 4g &\le 45 \\ 4g &\le 20 \\ g &\le 5 \end{aligned}

Priya can use at most 5 gigabytes. Check: $25 + 4(5) = 45$ , which is allowed. At $6$ gigabytes, the bill is $49$ dollars, which is too much.

Tip

Use the story to double-check the direction. A budget ("no more than") should give a maximum, so expect $\le$ . A goal ("at least") should give a minimum, so expect $\ge$ .