Unit 2 · Lesson 2.8

2.8Unit 2 Review: Inequalities

Bring together everything from Unit 2 — inequality symbols and graphs, one-step, two-step, and compound inequalities — in one comprehensive review and unit test.

Why This Matters

Inequalities are everywhere in the real world — from qualifying for a loan to meeting safety standards. Reviewing this unit solidifies the skills you'll need for systems of inequalities, linear programming, and the SAT.

Workbook

Lesson, vocabulary, worked examples, and practice problems.

Unit 2 Big Idea

An inequality describes a range of values, not just one. Solve it like an equation — but flip the symbol whenever you multiply or divide by a negative number. Graph the solution set and verify with a test value.

Guided Practice Video: Unit 2 Review

Watch the guided practice walkthrough for the Unit 2 review, then complete the problems below.

Video by Sang Real Math

Watch on YouTube ↗

Chapter-by-Chapter Summary

2.1

Introduction to Inequalities

Understand inequality symbols, graph solution sets on a number line, and write inequalities from real-world situations.

Symbols: < less than, > greater than, ≤ at most, ≥ at least

Open circle: < or > (endpoint not included)

Closed circle: ≤ or ≥ (endpoint included)

Example: x > 3 → open circle at 3, shade right

2.2

One-Step Inequalities

Apply one inverse operation. Flip the symbol only when multiplying or dividing by a negative.

Add/Subtract: x + 4 > 7 → x > 3 (no flip)

Multiply positive: 3x ≤ 12 → x ≤ 4 (no flip)

Divide negative: −2x < 8 → x > −4 (flip!)

Multiply negative: x/(−3) ≥ 2 → x ≤ −6 (flip!)

2.3

Two-Step Inequalities

Undo addition/subtraction first, then multiplication/division. Flip only at the second step, and only when the divisor/multiplier is negative.

Positive coeff: 2x + 3 > 11 → x > 4 (no flip)

Negative coeff: −3x + 6 > 0 → x < 2 (flip at step 2)

Fraction: x/2 − 4 ≥ −1 → x ≥ 6 (no flip)

Neg. fraction: x/(−4) + 2 ≥ 5 → x ≤ −12 (flip!)

2.4

Compound Inequalities

AND → intersection (segment); OR → union (two rays). Double inequalities are AND in three-part form.

AND: x > 1 AND x ≤ 5 → (1, 5]

OR: x < −2 OR x ≥ 4 → (−∞, −2) ∪ [4, ∞)

Double: −2 ≤ 2x + 4 < 10 → −3 ≤ x < 3

Neg. double: −6 < −2x ≤ 4 → −2 ≤ x < 3 (flip both!)

The Flip Rule — Master Reference

Symbol NEVER flips

Adding any number to both sides
Subtracting any number from both sides
Multiplying both sides by a positive
Dividing both sides by a positive

Symbol ALWAYS flips

Multiplying both sides by a negative
Dividing both sides by a negative
In a double inequality: flip both symbols

Interval Notation Quick Reference

x > a(a, ∞)

x ≥ a[a, ∞)

x < a(−∞, a)

x ≤ a(−∞, a]

a < x < b(a, b)

a ≤ x ≤ b[a, b]

a ≤ x < b[a, b)

a < x ≤ b(a, b]

x < a OR x > b(−∞, a) ∪ (b, ∞)

x ≤ a OR x ≥ b(−∞, a] ∪ [b, ∞)

No solution∅

All real numbers(−∞, ∞)

Mixed Review — By Chapter

Ch 01 — Introduction to Inequalities

Write an inequality for: "a number n is greater than −4".

Write an inequality for: "a number n is at most 7".

Graph x ≤ −2 on a number line. Use a closed or open circle?

Graph x > 5 on a number line. Use a closed or open circle?

Is x = 3 a solution of x ≥ 3? Is x = 2 a solution? Explain.

Write a real-world situation that can be modeled by x < 60.

Ch 02 — One-Step Inequalities

Solve and graph: x + 7 > 10

Solve and graph: x − 3 ≤ 5

Solve and graph: 4x < 20

Solve and graph: x/3 ≥ −2

Solve and graph: −5x > 15

Solve and graph: x/(−2) ≤ 4

Ch 03 — Two-Step Inequalities

Solve and graph: 2x + 5 > 13

Solve and graph: 3x − 4 ≤ 8

Solve and graph: −2x + 6 < 12

Solve and graph: −4x − 3 ≥ 9

Solve and graph: x/3 + 2 > 5

Solve and graph: x/(−2) − 1 ≤ 3

Ch 04 — Compound Inequalities

Solve and graph: x > −1 AND x ≤ 4

Solve and graph: x < −3 OR x ≥ 2

Solve and graph: 2x − 1 ≥ 3 AND 2x − 1 < 9

Solve and graph: 3x + 2 < −4 OR 3x + 2 ≥ 8

Solve the double inequality: −2 ≤ 2x + 4 < 10

Solve the double inequality: −9 < −3x − 3 ≤ 6

⚠️

Common Mistakes

❌

Forgetting to flip the inequality when multiplying or dividing by a negative number.

✅

This is the most common inequality error. Always flip the symbol when the multiplier/divisor is negative.

❌

Using a closed circle for strict inequalities (< or >) on the number line.

✅

Open circle for < and >. Closed circle (filled dot) for ≤ and ≥.

❌

Confusing 'AND' (intersection) with 'OR' (union) when graphing compound inequalities.

✅

'AND' shades the overlap region. 'OR' shades both outer regions.

❌

Writing interval notation with the wrong bracket type at the boundary.

✅

Parentheses for strict inequalities; brackets for ≤ or ≥. Infinity always uses a parenthesis.

Cumulative Mixed Practice

Solve and graph: −3x + 1 > 7. Write in interval notation.

Solve and graph: x/4 − 2 ≤ 1. Write in interval notation.

Solve and graph: 5x + 3 ≥ −12. Write in interval notation.

Solve and graph: −x − 4 < 2. Write in interval notation.

Solve and graph: 2x + 1 > 5 AND 2x + 1 ≤ 13. Write in interval notation.

Solve and graph: x − 3 < −1 OR x − 3 ≥ 4. Write in interval notation.

Solve: −1 ≤ 3x − 4 < 8. Write in interval notation.

Solve: −6 ≤ −2x + 2 < 4. Write in interval notation.

A student earns $12 per hour. She wants to earn more than $60. Write and solve an inequality for the number of hours h she must work.

The safe speed on a road is at least 25 mph and no more than 65 mph. Write a compound inequality for speed s and write the solution in interval notation.

A number is multiplied by −3 and then increased by 5. The result is less than 14. Write and solve the inequality.

Twice a number, decreased by 7, is at most 3. Write and solve the inequality.

Error Analysis

A student solved x + 5 > 3 and got x > 8. Find and correct the error.

A student solved −4x < 12 and got x < −3. Find and correct the error.

A student solved 2x + 3 ≤ 9 and got x ≤ 6. Find and correct the error.

A student solved −2x + 4 > 10 and got x > −3. Find and correct the error.

A student solved x > 5 AND x < 2 and wrote the solution as (2, 5). Find and correct the error.

A student solved −4 < −2x ≤ 8 and got −4 < x ≤ 4. Find and correct all errors.

Challenge Problems

Solve: 2(x + 3) > 10 AND 2(x + 3) ≤ 20. Write in interval notation.

Solve: −3(x − 2) < 9 OR −3(x − 2) ≥ 15. Write in interval notation.

Find all integers n such that −2n + 3 ≥ 7 and n > −5.

Write a two-step inequality whose solution is x ≤ −4 and that requires dividing by a negative number.

Write a compound AND inequality whose solution is [−3, 5) and that requires solving two separate inequalities.

2.7 · Graphing Linear Inequalities in Two Variables2.8 of 2.8Unit 2