← Algebra I

Algebra I

Solving Multi-Step Inequalities

Lesson

A multi-step inequality works just like a multi-step equation — distribute, combine like terms, then undo operations to isolate xx. The one rule that’s different: flip the inequality symbol when you multiply or divide both sides by a negative.

Order of attack:

  1. Distribute and combine like terms on each side, if needed.
  2. Undo addition or subtraction first.
  3. Undo multiplication or division — and flip if you divide or multiply by a negative.

Worked example 1

3x+4<193x + 4 < 19

Subtract 4:

3x<153x < 15

Divide by 3 (positive — no flip):

x<5x < 5

Worked example 2

−2x+5≤13-2x + 5 \le 13

Subtract 5 (no flip yet):

−2x≤8-2x \le 8

Divide by −2-2 — and flip the symbol:

x≥−4x \ge -4

How to type your answer

Use <, >, <=, or >=. Examples: x<5, x>=-4.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

2x+1<72x + 1 < 7

Problem 2

3x−4>53x - 4 > 5

Problem 3

x2+1≥4\frac{x}{2} + 1 \ge 4

Problem 4

4x+3≤114x + 3 \le 11

Practice

Standard problems matching the lesson.

Problem 5

3x+4<193x + 4 < 19

Problem 6

2x−7>52x - 7 > 5

Problem 7

5x+3≤285x + 3 \le 28

Problem 8

4x−9≥74x - 9 \ge 7

Problem 9

−2x+5≤13-2x + 5 \le 13

Problem 10

−3x−4<11-3x - 4 < 11

Problem 11

6−x>26 - x > 2

Problem 12

10−2x≥410 - 2x \ge 4

Problem 13

2(x+3)<142(x + 3) < 14

Problem 14

3(x−2)≥93(x - 2) \ge 9

Problem 15

5x+2<3x+105x + 2 < 3x + 10

Problem 16

−4(x+1)≥8-4(x + 1) \ge 8

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 17

5(x−1)>155(x - 1) > 15

Problem 18

3(x+2)≤123(x + 2) \le 12

Problem 19

−2(x−4)<6-2(x - 4) < 6

Problem 20

4x−7<2x+34x - 7 < 2x + 3

Problem 21

6−3x≥−96 - 3x \ge -9

Problem 22

−(x−4)>2-(x - 4) > 2

Practice

Standard problems matching the lesson.

Problem 23

A phone plan is 20 dollars base plus 1 dollar per minute. To stay under 50 dollars, solve 20 + x < 50.

Problem 24

A pool has 20 gallons and fills at 3 gallons per minute. To reach at least 50 gallons, solve 20 + 3x >= 50.

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 25

A taxi charges 5 dollars plus 2 dollars per mile. To spend at most 25 dollars, solve 5 + 2x <= 25.

Ask the tutor

Stuck on a concept? Want another example? Ask anything about this topic.

Type your own question below, or tap one of the suggestions. The tutor can re-explain the lesson, work through a specific problem with you, generate fresh practice tuned to where you are, or check your reasoning.

Quiz

Test yourself on this topic →

10 questions, no hints. About 5 minutes.