# 1.3 – Solving Linear Inequalities

## Objectives

• Solve multistep and compound inequalities.
• Graph compound Inequalities.
• Identify constraints in inequalities.

## Key Terms

• Compound Inequality – two or more simple inequalities joined together. Can be written as a chain or as separate statements joined together with “or” or “and.”
• Empty Set – a graph with No Solution, such as -2 > x > 12.  The graphs would not overlap.

## Notes

Notation for Apex Quizzes
• In Apex quizzes, use “<=” for and use “>=” for

Graphing on the Number Line
• When graphing inequalities, you may end up with two rays (OR), a line segment (AND), a line (All Real Numbers), an empty set – nothing to graph (No Solution), or even a single ray in one direction (AND, both greater or less than).
• Inequalities on the Number Line

• Compound Inequalities with “AND

• Compound Inequalities with “OR

## Examples

 Ex 1. What is the solution? Step 1: Write Inequality:  5x – 4 + 3x ≥ 38 – 2 Step 2: Combine like terms on EACH side of the inequality:  5x + 3x = 8x (on the left) and 38 – 2 = 36 (on the right) Step 3: Isolate “x” using inverse operations, starting with constants:  Add 4 on both sides(8x – 4 + 4 ≥ 36 + 4) Step 4: Isolate “x” using inverse operations, remove coefficients:  divide by 8 on both sides: $\frac{8x}{8}\geq\frac{40}{8}$ Step 5: Write your answer:  x ≥ 5 Ex 2. What is the solution? Step 1: Write Inequality: -2(5x + 1) ≥ 48 Step 2: Distribute -2 into the parenthesis: -10x – 1 ≥ 48 Step 3: Add 1 to both sides of the inequality:  -10x ≥ 49 Step 4: Divide both sides by -10 and FLIP the sign: $\frac{-10x}{-10}\leq\frac{49}{-10}$ Step 5: Write your answer:  x ≤ -4.9 Ex 3. A salesperson earns $300.50 per week plus 7% of her weekly sales. Which of the following describes the sales necessary for the salesperson to earn at least$900.85 in one week? Setup: weekly base plus commission of sales (x) equals her earnings 300.50 + 0.07x ≥ 900.85 x ≥ \$8576.43 We use the ≥ sign because she must earn “AT LEAST” that much.  She can earn more! Ex 4. Solve the compound inequality:-16 < 2x + 8 < -6 Step 1: Set up 2 inequalities -16 < 2x + 8 2x + 8 < -6 Step 2: Solve each inequality and graph them on a number line -12 < x x < -7 Ask yourself: do they overlap?  What is the final solution? Answer:  x > -7 (a ray, pointing to the right) Because BOTH inequalities contain everything greater than -7. Look carefully: only ONE of the inequalities contains the values smaller than -7 (-8, -9, -10, -11, etc.).