Skip to content
Mom's Basement University

Linear Inequalities: Multi-Step

What You’ll Learn

Section titled “What You’ll Learn”

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

The Concept

Section titled “The Concept”

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

3x+5>17Subtract 5 from both sides3x>12Divide both sides by 3x>4\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

−2x+8≤4Subtract 8 from both sides−2x≤−4Divide both sides by -2 (flip)x≥2\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≤5-3 < x + 2 \leq 5. Subtract 2 from all three parts: −5<x≤3-5 < x \leq 3

Worked Examples

Section titled “Worked Examples”

Solve and graph −4x+12≥8-4x + 12 \geq 8

−4x+12≥8Subtract 12 from both sides−4x≥−4Divide both sides by -4 (flip)x≤1\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≤1x \leq 1



Another: Solve 2(3x−1)<10+x2(3x - 1) < 10 + x

2(3x−1)<10+xDistribute6x−2<10+xSubtract x from both sides5x−2<10Add 2 to both sides5x<12Divide both sides by 5x<2.4\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.

Real-World Applications

Section titled “Real-World Applications”

Inequalities set ranges:

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

80+s≤300Subtract 80 from both sidess≤220\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?

22+h≥35Subtract 22 from both sidesh≥13\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?

80−80d<60Subtract 80 from both sides−80d<−20Divide both sides by -80 (flip)d>0.25\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%.

Quiz

Section titled “Quiz”
Solve 5x - 8 > 12.
Solve -3x + 6 ≤ 15.
You can spend at most 200 dollars. Which inequality?
Solve 2(4x - 3) > 10 + 6x.
Solve $-2(x + 4) \geq 6$.