Linear Inequalities: Multi-Step

In this lesson you’ll learn how to solve multi-step inequalities and graph their solutions on a number line.

Multi-step inequalities require several operations. Solve like equations, but:

• Flip the inequality sign when multiplying or dividing by a negative number
• Graph solutions: open circle for < or >, closed for ≤ or ≥, shade direction

Example: 3x + 5 > 17

$\begin{array}{l|rcl} & 3x + 5 &>& 17 \\ \text{Subtract 5 from both sides} & 3x &>& 12 \\ \text{Divide both sides by 3} & x &>& 4 \end{array}$

Graph: open circle at 4, shade right.

With negative: -2x + 8 ≤ 4

$\begin{array}{l|rcl} & -2x + 8 &\leq& 4 \\ \text{Subtract 8 from both sides} & -2x &\leq& -4 \\ \text{Divide both sides by -2 (flip)} & x &\geq& 2 \end{array}$

Graph: closed circle at 2, shade right.

Compound inequalities: $-3 < x + 2 \leq 5$. Subtract 2 from all three parts: $-5 < x \leq 3$

Solve and graph $-4x + 12 \geq 8$

$\begin{array}{l|rcl} & -4x + 12 &\geq& 8 \\ \text{Subtract 12 from both sides} & -4x &\geq& -4 \\ \text{Divide both sides by -4 (flip)} & x &\leq& 1 \end{array}$

Graph: closed circle at 1, shade left. All values where $x \leq 1$

Another: Solve $2(3x - 1) < 10 + x$

$\begin{array}{l|rcl} & 2(3x - 1) &<& 10 + x \\ \text{Distribute} & 6x - 2 &<& 10 + x \\ \text{Subtract x from both sides} & 5x - 2 &<& 10 \\ \text{Add 2 to both sides} & 5x &<& 12 \\ \text{Divide both sides by 5} & x &<& 2.4 \end{array}$

Graph: open circle at 2.4, shade left.

Inequalities set ranges:

Budget - You can spend at most 300 dollars total. You’ve already spent 80 dollars. How much more can you spend?

$\begin{array}{l|rcl} & 80 + s &\leq& 300 \\ \text{Subtract 80 from both sides} & s &\leq& 220 \end{array}$

You can spend up to 220 more dollars.

Overtime - You need at least 35 hours to qualify for overtime. You’ve worked 22 hours so far. How many more hours do you need?

$\begin{array}{l|rcl} & 22 + h &\geq& 35 \\ \text{Subtract 22 from both sides} & h &\geq& 13 \end{array}$

You need at least 13 more hours.

Sale discount - An item costs 80 dollars. You want to pay less than 60 dollars. What minimum discount percentage do you need?

$\begin{array}{l|rcl} & 80 - 80d &<& 60 \\ \text{Subtract 80 from both sides} & -80d &<& -20 \\ \text{Divide both sides by -80 (flip)} & d &>& 0.25 \end{array}$

You need a discount greater than 25%.