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Pre-Algebra

Solving Two-Step Inequalities

Lesson

A two-step inequality works just like a two-step equation: undo the addition or subtraction first, then undo the multiplication or division.

The same flip rule from one-step inequalities still applies.

The flip rule (still!)

Whenever you multiply or divide both sides by a negative number, you must flip the inequality symbol. << ↔ >>, ≤\le ↔ ≥\ge.

Adding or subtracting does NOT flip. Only multiplying or dividing by a negative.

Worked example 1 — no flip

2x+3<112x + 3 < 11

Subtract 3 from both sides:

2x<82x < 8

Divide both sides by 2:

x<4x < 4

Worked example 2 — with flip

−3x+2≥14-3x + 2 \ge 14

Subtract 2 from both sides:

−3x≥12-3x \ge 12

Divide both sides by −3-3. Because we divided by a negative, the inequality flips:

x≤−4x \le -4

How to type your answer

Write your answer in the form xx + symbol + number. Use <, >, <=, or >=. Examples: x<4, x>=-2.

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−2>103x - 2 > 10

Problem 3

4x+5≤134x + 5 \le 13

Problem 4

5x−3≥125x - 3 \ge 12

Practice

Standard problems matching the lesson.

Problem 5

2x+5<132x + 5 < 13

Problem 6

3x−4>143x - 4 > 14

Problem 7

4x+1≤94x + 1 \le 9

Problem 8

5x−8≥125x - 8 \ge 12

Problem 9

x2+3>7\frac{x}{2} + 3 > 7

Problem 10

x3−1<4\frac{x}{3} - 1 < 4

Problem 11

6x+7≤−56x + 7 \le -5

Problem 12

−2x+9>1-2x + 9 > 1

Problem 13

−x+10≥3-x + 10 \ge 3

Problem 14

2x−5≤−112x - 5 \le -11

Problem 15

x4+2≥−1\frac{x}{4} + 2 \ge -1

Problem 16

3x+1<73x + 1 < 7

Problem 17

5x+20<605x + 20 < 60

Problem 18

2x+4≤162x + 4 \le 16

Challenge

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

Problem 19

−3x+8≥−7-3x + 8 \ge -7

Problem 20

4x−9<−254x - 9 < -25

Problem 21

x5+7>4\frac{x}{5} + 7 > 4

Problem 22

−2x−5≤7-2x - 5 \le 7

Problem 23

5x−11<−15x - 11 < -1

Problem 24

x−3+2≥6\frac{x}{-3} + 2 \ge 6

Problem 25

15x+50≤20015x + 50 \le 200

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