Inequalities · One-step · 2.3

The flip rule.

The one rule inequalities add to equations: when you multiply or divide BOTH sides by a NEGATIVE number, you must REVERSE the inequality symbol. −2x < 8 → divide by −2 and flip → x > −4. (Check: x = 0 gives 0 < 8, true, and 0 > −4, true.)

01 / THE IDEA

See it work.

Watch the method

The flip rule

−2x < 8 → ÷(−2), FLIP → x > −4.

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−2x<8-2x < 8
1

The inequality

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

−2x<8-2x < 8
2

Flip the sign!

Dividing both sides by the negative -2 reverses the inequality.

x>−4x > -4

On the number line

Shade the solutions.

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

-9-8-7-6-5-4-3-2-101
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.

Why it flips

Multiplying by a negative reverses order on the number line (3 > 2 but −3 < −2), so the symbol must flip.

Why it works

Negatives turn the number line around.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Solve for x: -3x ≥ -3

1

The inequality

Solve for x just like an equation — with one twist. -3x ≥ -3

2

Flip the sign!

Dividing both sides by the negative -3 reverses the inequality. x ≤ 1

EX 2medium

Solve for x: -4x ≤ -20

1

The inequality

Solve for x just like an equation — with one twist. -4x ≤ -20

2

Flip the sign!

Dividing both sides by the negative -4 reverses the inequality. x ≥ 5

EX 3hard

Solve for x: -4x > -24

1

The inequality

Solve for x just like an equation — with one twist. -4x > -24

2

Flip the sign!

Dividing both sides by the negative -4 reverses the inequality. x < 6

04 / COMMON TRAPS

Where students slip.

Trap

Forgetting to flip.

Fix

A negative coefficient ALWAYS flips the symbol.

Trap

Flipping for a negative on the other side.

Fix

Only flip when ×/÷ by a negative — not for moving a negative term.

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 ≥ 15

Need a hint?
  • Divide by the negative coefficient.
  • Then reverse the inequality sign.
C2

Solve for x: -6x ≥ -36

Need a hint?
  • Divide by the negative coefficient.
  • Then reverse the inequality sign.
C3

Solve for x: -6x ≥ 12

Need a hint?
  • Divide by the negative coefficient.
  • Then reverse the inequality sign.
C4

Solve for x: -5x ≤ 20

Need a hint?
  • Divide by the negative coefficient.
  • Then reverse the inequality sign.
Apply

Apply (4)

A1

Solve for x: -6x ≤ -6

Need a hint?
  • Divide by the negative coefficient.
  • Then reverse the inequality sign.
A2

Solve for x: -3x ≥ -9

Need a hint?
  • Divide by the negative coefficient.
  • Then reverse the inequality sign.
A3

Solve for x: -6x > -6

Need a hint?
  • Divide by the negative coefficient.
  • Then reverse the inequality sign.
A4

Solve for x: -5x ≥ 5

Need a hint?
  • Divide by the negative coefficient.
  • Then reverse the inequality sign.
Extension

Extension (4)

E1

Solve for x: -3x ≤ 9

Need a hint?
  • Divide by the negative coefficient.
  • Then reverse the inequality sign.
E2

Solve for x: -5x > -25

Need a hint?
  • Divide by the negative coefficient.
  • Then reverse the inequality sign.
E3

Solve for x: -4x ≥ 12

Need a hint?
  • Divide by the negative coefficient.
  • Then reverse the inequality sign.
E4

Solve for x: -6x ≥ -6

Need a hint?
  • Divide by the negative coefficient.
  • Then reverse the inequality sign.