Inequalities · Capstone · C1

Solve it all.

Everything together: translate words, graph solutions, and solve one-step, two-step, both-sides, and distribute inequalities — always reversing the sign when you multiply or divide by a negative.

01 / THE IDEA

See it work.

Watch the method

Solve it all

Read the problem, solve carefully, and watch the flip.

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−3x+2<11-3x + 2 < 11
1

The inequality

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

−3x+2<11-3x + 2 < 11
2

Isolate the x-term

Undo the +/− so only the x-term is left.

−3x<9-3x < 9
3

Flip the sign!

Dividing both sides by the negative -3 reverses the inequality.

x>−3x > -3

On the number line

Shade the solutions.

A open circle excludes -3; the arrow points to every value that works.

-8-7-6-5-4-3-2-1012
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 toolkit

Distribute, gather, divide; flip on a negative; graph with open/closed circles.

Why it works

One rule sets inequalities apart — the flip.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Solve for x: 5x-4 ≤ 4x-1

1

The inequality

Solve for x just like an equation — with one twist. 5x - 4 ≤ 4x - 1

2

Isolate x

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

EX 2medium

Solve for x: -6x > -12

1

The inequality

Solve for x just like an equation — with one twist. -6x > -12

2

Flip the sign!

Dividing both sides by the negative -6 reverses the inequality. x < 2

EX 3hard

Solve for x: 6x+4 ≤ 2x-12

1

The inequality

Solve for x just like an equation — with one twist. 6x + 4 ≤ 2x - 12

2

Isolate x

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

04 / COMMON TRAPS

Where students slip.

Trap

Forgetting the flip on a negative.

Fix

Always reverse the sign when ×/÷ by a negative.

Trap

Open/closed circle mix-ups.

Fix

Strict → open; inclusive → closed.

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+1 ≤ 3x+7

Need a hint?
  • Solve like an equation.
  • Flip the sign on a negative ×/÷.
C2

Solve for x: -3x < -3

Need a hint?
  • Solve like an equation.
  • Flip the sign on a negative ×/÷.
C3

Solve for x: 5x+5 < 1x+13

Need a hint?
  • Solve like an equation.
  • Flip the sign on a negative ×/÷.
C4

Solve for x: -3x+6 ≥ 9

Need a hint?
  • Solve like an equation.
  • Flip the sign on a negative ×/÷.
Apply

Apply (4)

A1

Solve for x: x-3 > 6

Need a hint?
  • Solve like an equation.
  • Flip the sign on a negative ×/÷.
A2

Solve for x: 5x-6 ≥ 3x-14

Need a hint?
  • Solve like an equation.
  • Flip the sign on a negative ×/÷.
A3

Solve for x: x+6 ≤ -8

Need a hint?
  • Solve like an equation.
  • Flip the sign on a negative ×/÷.
A4

Solve for x: x+9 < -3

Need a hint?
  • Solve like an equation.
  • Flip the sign on a negative ×/÷.
Extension

Extension (4)

E1

Solve for x: -2x-6 < -10

Need a hint?
  • Solve like an equation.
  • Flip the sign on a negative ×/÷.
E2

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

Need a hint?
  • Solve like an equation.
  • Flip the sign on a negative ×/÷.
E3

Solve for x: -6x < 36

Need a hint?
  • Solve like an equation.
  • Flip the sign on a negative ×/÷.
E4

Solve for x: 3x+1 ≥ 1x+3

Need a hint?
  • Solve like an equation.
  • Flip the sign on a negative ×/÷.