Inequalities · Apply · 4.2

Solve a real constraint.

Many word problems set up as a two-step inequality: a fixed amount plus a per-unit cost must stay within a limit. “A $20 fee plus $5 per ticket, budget $60”: 5x + 20 ≤ 60 → x ≤ 8, so at most 8 tickets.

01 / THE IDEA

See it work.

Watch the method

Solve a real constraint

5x + 20 ≤ 60 → x ≤ 8 (at most 8).

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5x+20≤605x + 20 \le 60
1

The inequality

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

5x+20≤605x + 20 \le 60
2

Isolate x

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

x≤8x ≤ 8

On the number line

Shade the solutions.

A closed circle includes 8; the arrow points to every value that works.

345678910111213
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.

Model & solve

Write fixed + rate·x compared to the limit, solve, and read the answer in context.

Why it works

How many fit under the cap?

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Solve for x: -3x-2 > -14

1

The inequality

Solve for x just like an equation — with one twist. -3x - 2 > -14

2

Isolate the x-term

Undo the +/− so only the x-term is left. -3x > -12

3

Flip the sign!

Dividing both sides by the negative -3 reverses the inequality. x < 4

EX 2medium

Solve for x: -4x+1 ≥ 13

1

The inequality

Solve for x just like an equation — with one twist. -4x + 1 ≥ 13

2

Isolate the x-term

Undo the +/− so only the x-term is left. -4x ≥ 12

3

Flip the sign!

Dividing both sides by the negative -4 reverses the inequality. x ≤ -3

EX 3hard

Solve for x: 5x+8 > 13

1

The inequality

Solve for x just like an equation — with one twist. 5x + 8 > 13

2

Isolate x

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

04 / COMMON TRAPS

Where students slip.

Trap

Using = instead of ≤.

Fix

A budget is a limit, so use ≤ (or ≥).

Trap

Forgetting to interpret.

Fix

Translate x ≤ 8 back to “at most 8”.

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: -1x+4 ≤ 2

Need a hint?
  • Write fixed + rate·x ≤ limit.
  • Solve, then state it in words.
C2

Solve for x: 2x-1 > -11

Need a hint?
  • Write fixed + rate·x ≤ limit.
  • Solve, then state it in words.
C3

Solve for x: -2x+4 ≥ 10

Need a hint?
  • Write fixed + rate·x ≤ limit.
  • Solve, then state it in words.
C4

Solve for x: 4x+2 > -2

Need a hint?
  • Write fixed + rate·x ≤ limit.
  • Solve, then state it in words.
Apply

Apply (4)

A1

Solve for x: -2x-6 ≤ 0

Need a hint?
  • Write fixed + rate·x ≤ limit.
  • Solve, then state it in words.
A2

Solve for x: 1x+8 ≤ 9

Need a hint?
  • Write fixed + rate·x ≤ limit.
  • Solve, then state it in words.
A3

Solve for x: 3x-3 ≥ 3

Need a hint?
  • Write fixed + rate·x ≤ limit.
  • Solve, then state it in words.
A4

Solve for x: -1x+3 ≥ 0

Need a hint?
  • Write fixed + rate·x ≤ limit.
  • Solve, then state it in words.
Extension

Extension (4)

E1

Solve for x: -2x-3 ≥ -7

Need a hint?
  • Write fixed + rate·x ≤ limit.
  • Solve, then state it in words.
E2

Solve for x: -3x+8 < 14

Need a hint?
  • Write fixed + rate·x ≤ limit.
  • Solve, then state it in words.
E3

Solve for x: 1x+4 < 0

Need a hint?
  • Write fixed + rate·x ≤ limit.
  • Solve, then state it in words.
E4

Solve for x: -2x+8 < 12

Need a hint?
  • Write fixed + rate·x ≤ limit.
  • Solve, then state it in words.