Inequalities · Multi-step · 3.2

Variables on both sides.

When x appears on both sides, move all the x-terms to one side and the numbers to the other, then finish as a one- or two-step. 5x + 1 > 2x + 7 → 3x > 6 → x > 2.

01 / THE IDEA

See it work.

Watch the method

Variables on both sides

5x + 1 > 2x + 7 → 3x > 6 → x > 2.

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5x+1>2x+75x + 1 > 2x + 7
1

The inequality

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

5x+1>2x+75x + 1 > 2x + 7
2

Isolate x

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

x>2x > 2
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.

Gather like terms

Subtract the smaller x-term from both sides so x lives on one side only.

Why it works

Get all the x’s together first.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Solve for x: 4x+6 ≤ 1x+9

1

The inequality

Solve for x just like an equation — with one twist. 4x + 6 ≤ 1x + 9

2

Isolate x

Undo the operations to get x alone (the sign stays the same). x ≤ 1

EX 2medium

Solve for x: 4x+1 ≥ 1x+10

1

The inequality

Solve for x just like an equation — with one twist. 4x + 1 ≥ 1x + 10

2

Isolate x

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

EX 3hard

Solve for x: 6x+6 > 4x+4

1

The inequality

Solve for x just like an equation — with one twist. 6x + 6 > 4x + 4

2

Isolate x

Undo the operations to get x alone (the sign stays the same). x > -1

04 / COMMON TRAPS

Where students slip.

Trap

Leaving x on both sides.

Fix

Move one x-term across so x is on a single side.

Trap

Sign slips moving terms.

Fix

Subtract the same term from both sides.

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-4 < 3x-6

Need a hint?
  • Subtract the smaller x-term from both sides.
  • Then solve the two-step that remains.
C2

Solve for x: 4x+2 ≤ 2x-2

Need a hint?
  • Subtract the smaller x-term from both sides.
  • Then solve the two-step that remains.
C3

Solve for x: 5x+1 > 4x+6

Need a hint?
  • Subtract the smaller x-term from both sides.
  • Then solve the two-step that remains.
C4

Solve for x: 4x-1 ≥ 3x-3

Need a hint?
  • Subtract the smaller x-term from both sides.
  • Then solve the two-step that remains.
Apply

Apply (4)

A1

Solve for x: 5x-3 < 4x+2

Need a hint?
  • Subtract the smaller x-term from both sides.
  • Then solve the two-step that remains.
A2

Solve for x: 3x+6 ≥ 2x+11

Need a hint?
  • Subtract the smaller x-term from both sides.
  • Then solve the two-step that remains.
A3

Solve for x: 5x+5 < 3x+11

Need a hint?
  • Subtract the smaller x-term from both sides.
  • Then solve the two-step that remains.
A4

Solve for x: 3x+6 < 1x+16

Need a hint?
  • Subtract the smaller x-term from both sides.
  • Then solve the two-step that remains.
Extension

Extension (4)

E1

Solve for x: 4x+1 < 3x-2

Need a hint?
  • Subtract the smaller x-term from both sides.
  • Then solve the two-step that remains.
E2

Solve for x: 6x+2 ≥ 3x+17

Need a hint?
  • Subtract the smaller x-term from both sides.
  • Then solve the two-step that remains.
E3

Solve for x: 6x-6 < 4x-2

Need a hint?
  • Subtract the smaller x-term from both sides.
  • Then solve the two-step that remains.
E4

Solve for x: 5x+1 ≤ 1x+17

Need a hint?
  • Subtract the smaller x-term from both sides.
  • Then solve the two-step that remains.