Inequalities · Multi-step · 3.4

Multi-step mastery.

Mixed practice: distribute if needed, gather x’s on one side and numbers on the other, then divide — flipping the sign only when you divide by a negative. Same toolkit, harder problems.

01 / THE IDEA

See it work.

Watch the method

Multi-step mastery

Distribute, gather, divide (flip on a negative).

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4x−1≥x+84x - 1 \ge x + 8
1

The inequality

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

4x−1≥x+84x - 1 \ge x + 8
2

Isolate x

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

x≥3x ≥ 3
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.

The full routine

Open brackets, collect like terms, then isolate x with the flip in mind.

Why it works

One careful pass, top to bottom.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Solve for x: -6x ≥ 30

1

The inequality

Solve for x just like an equation — with one twist. -6x ≥ 30

2

Flip the sign!

Dividing both sides by the negative -6 reverses the inequality. x ≤ -5

EX 2medium

Solve for x: -6x > -30

1

The inequality

Solve for x just like an equation — with one twist. -6x > -30

2

Flip the sign!

Dividing both sides by the negative -6 reverses the inequality. x < 5

EX 3hard

Solve for x: -5x ≤ 10

1

The inequality

Solve for x just like an equation — with one twist. -5x ≤ 10

2

Flip the sign!

Dividing both sides by the negative -5 reverses the inequality. x ≥ -2

04 / COMMON TRAPS

Where students slip.

Trap

Skipping a step under pressure.

Fix

Work in order: distribute, gather, divide.

Trap

Missing the flip.

Fix

Check the sign of the coefficient before dividing.

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: -5x > -30

Need a hint?
  • Distribute, then gather like terms.
  • Divide last — flip if negative.
C2

Solve for x: 6x+6 < 3x+3

Need a hint?
  • Distribute, then gather like terms.
  • Divide last — flip if negative.
C3

Solve for x: 3x+1 ≥ 1x+11

Need a hint?
  • Distribute, then gather like terms.
  • Divide last — flip if negative.
C4

Solve for x: -5x < -5

Need a hint?
  • Distribute, then gather like terms.
  • Divide last — flip if negative.
Apply

Apply (4)

A1

Solve for x: -6x ≥ 30

Need a hint?
  • Distribute, then gather like terms.
  • Divide last — flip if negative.
A2

Solve for x: 6x-6 ≤ 3x-3

Need a hint?
  • Distribute, then gather like terms.
  • Divide last — flip if negative.
A3

Solve for x: 3x-1 < 1x+3

Need a hint?
  • Distribute, then gather like terms.
  • Divide last — flip if negative.
A4

Solve for x: 5x+8 < -12

Need a hint?
  • Distribute, then gather like terms.
  • Divide last — flip if negative.
Extension

Extension (4)

E1

Solve for x: 5x-6 < 2x-18

Need a hint?
  • Distribute, then gather like terms.
  • Divide last — flip if negative.
E2

Solve for x: x-7 < -9

Need a hint?
  • Distribute, then gather like terms.
  • Divide last — flip if negative.
E3

Solve for x: 1x-3 < -6

Need a hint?
  • Distribute, then gather like terms.
  • Divide last — flip if negative.
E4

Solve for x: 5x-7 ≥ -2

Need a hint?
  • Distribute, then gather like terms.
  • Divide last — flip if negative.