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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

Alg2A 1.03 - Inequalities on Line


 

  • Compound Inequalities with “AND

Alg2A 1.03 - Compound Inequalities AND


 

  • Compound Inequalities with “OR

Alg2A 1.03 - Compound Inequalities 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?

Alg2A 1.03 - Inequality Graph 02

Alg2A 1.03 - Inequality Graph 01

  • 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.).

 


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