Inequalities · Multi-step · 3.3

Distribute first.

If a bracket multiplies a group, distribute it before isolating x. 3(x − 2) ≥ 6 → 3x − 6 ≥ 6 → 3x ≥ 12 → x ≥ 4.

01 / THE IDEA

See it work.

Watch the method

Distribute first

3(x − 2) ≥ 6 → 3x − 6 ≥ 6 → x ≥ 4.

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3(x−2)≥63(x - 2) \ge 6
1

The inequality

Solve for x just like an equation — with one twist.

3(x−2)≥63(x - 2) \ge 6
2

Isolate x

Undo the operations to get x alone (the sign stays the same).

x≥4x ≥ 4
02 / YOUR TURN

Practice until it feels automatic.

Your turn

Solve for x.

Type the solution, e.g. x < 4 (you can write <= for ≤).

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— / WHY IT WORKS

The idea underneath.

Clear the brackets

Multiply each term inside the brackets, then proceed as usual.

Why it works

Open it up before isolating x.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Solve for x: 3(x+2) ≥ -6

1

The inequality

Solve for x just like an equation — with one twist. 3(x + 2) ≥ -6

2

Isolate x

Undo the operations to get x alone (the sign stays the same). x ≥ -4

EX 2medium

Solve for x: 3(x+1) < 12

1

The inequality

Solve for x just like an equation — with one twist. 3(x + 1) < 12

2

Isolate x

Undo the operations to get x alone (the sign stays the same). x < 3

EX 3hard

Solve for x: 2(x+3) ≤ 10

1

The inequality

Solve for x just like an equation — with one twist. 2(x + 3) ≤ 10

2

Isolate x

Undo the operations to get x alone (the sign stays the same). x ≤ 2

04 / COMMON TRAPS

Where students slip.

Trap

Multiplying only the first term.

Fix

Distribute to EVERY term inside.

Trap

Dropping a sign inside.

Fix

Keep the signs as you distribute.

05 / QUIZ

Test yourself.

The Test

12 problems across three tiers. Auto-graded on submit. Hints and a full walkthrough on every question. Your work is saved on this device — sign in to keep it across devices.

0/12Answered
How to answerType just the value. Formatting won't trip you up — we ignore spaces, capital letters, commas, and subscript style, so t₁ = 5, d = 3 and t1=5,d=3 both count. For a list of numbers, separate them with commas (e.g. 3, 10, 17).
Core

Core (4)

C1

Solve for x: 2(x-5) > -8

Need a hint?
  • Multiply through the bracket.
  • Then solve the two-step.
C2

Solve for x: 3(x+1) ≥ 6

Need a hint?
  • Multiply through the bracket.
  • Then solve the two-step.
C3

Solve for x: 2(x-5) ≥ -16

Need a hint?
  • Multiply through the bracket.
  • Then solve the two-step.
C4

Solve for x: 2(x-5) < -8

Need a hint?
  • Multiply through the bracket.
  • Then solve the two-step.
Apply

Apply (4)

A1

Solve for x: 3(x-2) > -18

Need a hint?
  • Multiply through the bracket.
  • Then solve the two-step.
A2

Solve for x: 3(x+1) < 18

Need a hint?
  • Multiply through the bracket.
  • Then solve the two-step.
A3

Solve for x: 3(x+3) > 21

Need a hint?
  • Multiply through the bracket.
  • Then solve the two-step.
A4

Solve for x: 2(x-3) < -8

Need a hint?
  • Multiply through the bracket.
  • Then solve the two-step.
Extension

Extension (4)

E1

Solve for x: 3(x+1) < 6

Need a hint?
  • Multiply through the bracket.
  • Then solve the two-step.
E2

Solve for x: 3(x-4) < -18

Need a hint?
  • Multiply through the bracket.
  • Then solve the two-step.
E3

Solve for x: 2(x-3) ≤ 2

Need a hint?
  • Multiply through the bracket.
  • Then solve the two-step.
E4

Solve for x: 4(x+5) ≤ 12

Need a hint?
  • Multiply through the bracket.
  • Then solve the two-step.