Grade: 9 Subject: Math (Algebra I) Unit: Linear Equations & Inequalities Lesson: 2 of 6 SAT: Algebra ACT: Math

Linear Inequalities

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Linear Inequality

A linear inequality is a mathematical statement that compares two expressions using inequality symbols: < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). Unlike equations that have one solution, inequalities typically have infinitely many solutions.

Linear inequalities are solved using the same techniques as linear equations, with one critical difference: when you multiply or divide both sides by a negative number, you must flip the inequality symbol.

Inequality Symbols and Their Meanings

Symbol Meaning Example Graph Notation
< Less than x < 5 Open circle, shade left
> Greater than x > 3 Open circle, shade right
≤ Less than or equal to x ≤ 7 Closed circle, shade left
≥ Greater than or equal to x ≥ -2 Closed circle, shade right

The Critical Rule: Flipping the Inequality

When you multiply or divide both sides of an inequality by a negative number, you must reverse (flip) the inequality sign. This is because multiplying by a negative reverses the order of numbers on the number line.

Example: If -2x > 6, dividing by -2 gives x < -3 (not x > -3)

Solving Steps for Linear Inequalities

  1. Simplify each side (distribute, combine like terms)
  2. Collect variable terms on one side
  3. Collect constants on the other side
  4. Isolate the variable (remember to flip if dividing by negative)
  5. Express the solution and/or graph it on a number line

Interval Notation

Inequality Interval Notation Description
x > 3 (3, infinity) All numbers greater than 3
x ≥ 3 [3, infinity) All numbers 3 or greater
x < 5 (-infinity, 5) All numbers less than 5
x ≤ 5 (-infinity, 5] All numbers 5 or less

SAT/ACT Connection

Linear inequalities appear frequently on standardized tests, often in word problem contexts like "at most," "at least," "no more than," and "no fewer than." Understanding how to translate these phrases into inequality symbols is essential for test success.

Examples

Work through these examples to see how to solve linear inequalities step by step.

Example 1: Basic Inequality

Problem: Solve 3x + 5 > 14

Step 1: Subtract 5 from both sides

3x > 9

Step 2: Divide both sides by 3 (positive, so no flip)

x > 3

Solution: x > 3, or in interval notation: (3, infinity)

Graph: Open circle at 3, shade to the right

Example 2: Dividing by a Negative

Problem: Solve -4x + 2 ≤ 18

Step 1: Subtract 2 from both sides

-4x ≤ 16

Step 2: Divide both sides by -4 (negative, so FLIP the sign)

x ≥ -4

Solution: x ≥ -4, or in interval notation: [-4, infinity)

Graph: Closed circle at -4, shade to the right

Example 3: Inequality with Distributive Property

Problem: Solve 2(x - 3) < 4x + 2

Step 1: Distribute the 2

2x - 6 < 4x + 2

Step 2: Subtract 2x from both sides

-6 < 2x + 2

Step 3: Subtract 2 from both sides

-8 < 2x

Step 4: Divide both sides by 2

-4 < x, which means x > -4

Solution: x > -4, or in interval notation: (-4, infinity)

Example 4: Compound Inequality

Problem: Solve -3 < 2x + 1 ≤ 7

Step 1: Subtract 1 from all three parts

-4 < 2x ≤ 6

Step 2: Divide all parts by 2

-2 < x ≤ 3

Solution: -2 < x ≤ 3, or in interval notation: (-2, 3]

Graph: Open circle at -2, closed circle at 3, shade between

Example 5: Word Problem

Problem: A phone plan costs $30 per month plus $0.05 per text message. If you want to spend at most $50 per month, how many text messages can you send?

Step 1: Set up the inequality (let t = number of texts)

30 + 0.05t ≤ 50

Step 2: Subtract 30 from both sides

0.05t ≤ 20

Step 3: Divide both sides by 0.05

t ≤ 400

Solution: You can send at most 400 text messages per month.

Practice

Solve each inequality and select the correct answer.

1. Solve: x + 7 > 12

A) x > 5   B) x > 19   C) x < 5   D) x < 19

2. Solve: 4x - 3 ≤ 13

A) x ≤ 4   B) x ≥ 4   C) x ≤ 2.5   D) x ≥ 2.5

3. Solve: -2x > 10

A) x > 5   B) x > -5   C) x < 5   D) x < -5

4. Solve: 3(x + 2) ≥ 15

A) x ≥ 3   B) x ≤ 3   C) x ≥ 7   D) x ≤ 7

5. Solve: 5 - 2x < 11

A) x > -3   B) x < -3   C) x > 3   D) x < 3

6. Solve: 2x + 5 > x - 3

A) x > -8   B) x < -8   C) x > 8   D) x < 8

7. Solve: -3(x - 4) ≤ 6

A) x ≥ 2   B) x ≤ 2   C) x ≥ -2   D) x ≤ -2

8. Solve: (x + 4)/2 > 5

A) x > 6   B) x > 3   C) x < 6   D) x < 3

9. What is the interval notation for x ≥ -3?

A) (-3, infinity)   B) [-3, infinity)   C) (-infinity, -3)   D) (-infinity, -3]

10. Maria earns $12 per hour. She wants to earn at least $180. What is the minimum hours she must work?

A) h ≥ 15   B) h ≤ 15   C) h > 15   D) h < 15

Click to reveal answers
  1. A) x > 5 - Subtract 7: x > 5
  2. A) x ≤ 4 - Add 3: 4x ≤ 16; divide by 4: x ≤ 4
  3. D) x < -5 - Divide by -2 and flip: x < -5
  4. A) x ≥ 3 - Distribute: 3x + 6 ≥ 15; subtract 6: 3x ≥ 9; divide: x ≥ 3
  5. A) x > -3 - Subtract 5: -2x < 6; divide by -2 and flip: x > -3
  6. A) x > -8 - Subtract x: x + 5 > -3; subtract 5: x > -8
  7. A) x ≥ 2 - Distribute: -3x + 12 ≤ 6; subtract 12: -3x ≤ -6; divide by -3 and flip: x ≥ 2
  8. A) x > 6 - Multiply by 2: x + 4 > 10; subtract 4: x > 6
  9. B) [-3, infinity) - Bracket includes -3, parenthesis at infinity
  10. A) h ≥ 15 - 12h ≥ 180; h ≥ 15 hours

Check Your Understanding

Answer these reflection questions to deepen your understanding.

1. Why must you flip the inequality sign when multiplying or dividing by a negative number?

Reveal Answer

Multiplying or dividing by a negative number reverses the order of numbers on the number line. For example, 2 < 5 is true, but if we multiply both sides by -1, we get -2 and -5, and now -2 > -5. The relationship has reversed, so we must flip the inequality sign to maintain a true statement.

2. What is the difference between an open circle and a closed circle when graphing inequalities?

Reveal Answer

An open circle indicates that the endpoint is NOT included in the solution (used for < and >). A closed (filled) circle indicates that the endpoint IS included in the solution (used for ≤ and ≥). In interval notation, this corresponds to using parentheses (exclude) versus brackets (include).

3. How do you translate "at most" and "at least" into mathematical symbols?

Reveal Answer

"At most" means the maximum value allowed, so it translates to ≤ (less than or equal to). "At least" means the minimum value required, so it translates to ≥ (greater than or equal to). For example, "You can spend at most $50" becomes "spending ≤ 50" and "You need at least 10 points" becomes "points ≥ 10."

4. Why do linear inequalities have infinitely many solutions while linear equations typically have exactly one solution?

Reveal Answer

A linear equation asks "what value makes both sides equal?" - there's usually exactly one such value. An inequality asks "what values make one side greater (or less) than the other?" - there are infinitely many such values. For example, x = 3 has one solution, but x > 3 includes 3.1, 4, 5, 100, and infinitely many other numbers.

🚀 Next Steps

  • Review any concepts that felt challenging
  • Move on to the next lesson when ready
  • Return to practice problems periodically for review