Inequalities · Multi-step · 3.1

Two-step inequalities.

Two-step inequalities undo operations in reverse order: clear the added constant first, then divide by the coefficient (flipping if it’s negative). 2x + 3 < 11 → 2x < 8 → x < 4.

01 / THE IDEA

See it work.

Watch the method

Two-step inequalities

2x + 3 < 11 → 2x < 8 → x < 4.

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2x+3<112x + 3 < 11
1

The inequality

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

2x+3<112x + 3 < 11
2

Isolate x

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

x<4x < 4

On the number line

Shade the solutions.

A open circle excludes 4; the arrow points to every value that works.

-10123456789
02 / YOUR TURN

Practice until it feels automatic.

Your turn

Solve for x.

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

streak 0
best 0
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— / WHY IT WORKS

The idea underneath.

Reverse order

Peel off the constant, then the coefficient — the same order as solving an equation.

Why it works

Last on, first off.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Solve for x: 2x-1 > -3

1

The inequality

Solve for x just like an equation — with one twist. 2x - 1 > -3

2

Isolate x

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

EX 2medium

Solve for x: 1x+6 ≥ 11

1

The inequality

Solve for x just like an equation — with one twist. 1x + 6 ≥ 11

2

Isolate x

Undo the operations to get x alone (the sign stays the same). x ≥ 5

EX 3hard

Solve for x: 3x+4 < 19

1

The inequality

Solve for x just like an equation — with one twist. 3x + 4 < 19

2

Isolate x

Undo the operations to get x alone (the sign stays the same). x < 5

04 / COMMON TRAPS

Where students slip.

Trap

Dividing before subtracting.

Fix

Clear the constant first.

Trap

Forgetting the flip on a negative coefficient.

Fix

If the coefficient is negative, reverse the sign.

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: -3x-7 ≤ -1

Need a hint?
  • Subtract the constant first.
  • Then divide (flip if negative).
C2

Solve for x: -3x+5 ≥ -4

Need a hint?
  • Subtract the constant first.
  • Then divide (flip if negative).
C3

Solve for x: 4x-3 ≤ 1

Need a hint?
  • Subtract the constant first.
  • Then divide (flip if negative).
C4

Solve for x: 5x+3 ≤ 8

Need a hint?
  • Subtract the constant first.
  • Then divide (flip if negative).
Apply

Apply (4)

A1

Solve for x: -1x+3 ≥ 6

Need a hint?
  • Subtract the constant first.
  • Then divide (flip if negative).
A2

Solve for x: -1x-3 ≤ -6

Need a hint?
  • Subtract the constant first.
  • Then divide (flip if negative).
A3

Solve for x: 3x-3 ≤ 6

Need a hint?
  • Subtract the constant first.
  • Then divide (flip if negative).
A4

Solve for x: -1x+4 ≥ 1

Need a hint?
  • Subtract the constant first.
  • Then divide (flip if negative).
Extension

Extension (4)

E1

Solve for x: 5x+2 ≥ 12

Need a hint?
  • Subtract the constant first.
  • Then divide (flip if negative).
E2

Solve for x: 4x+3 < 11

Need a hint?
  • Subtract the constant first.
  • Then divide (flip if negative).
E3

Solve for x: 2x+2 ≥ 0

Need a hint?
  • Subtract the constant first.
  • Then divide (flip if negative).
E4

Solve for x: -2x-3 ≤ 7

Need a hint?
  • Subtract the constant first.
  • Then divide (flip if negative).