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1.7 – Solving Inequalities with Multiplication and Division

Objectives

  • Isolate variables by reversing the operation found in an inequality.
  • Solve inequalities involving multiplication and division by using the properties of inequalities.
  • Ask the necessary questions to turn real-life problems into mathematical sentences.

 

Key Terms

(No key terms this section)

 

Notes

  • Rules
    • When you multiply or divide by a negative, flip the sign
    • If a < b, and c is positive, then ac < bc and \frac{a}{c}\textless\frac{b}{c}
      • Example: If 3 < 5, and 4 is positive, then (3)(4) < (5)(4) and \frac{3}{4}\textless\frac{5}{4}
        • Because 12 < 20 (12 is to the left of 20 on the number line; left is less)
        • Because 3/4 < 5/4 (in decimal form is: 0.75 is to the left of 1.25 on the number line; left is less)
    • If a < b, and c is negative, then ac > bc and \frac{a}{c}\textgreater\frac{b}{c}
      • Example: If 3 < 5, and 4 is negative, then (3)(-4) > (5)(-4) and \frac{3}{-4}\textgreater\frac{5}{-4}
        • Because -12 > -20 (-12 is to the right of -20 on the number line; right is greater)
        • Because -3/4 > -5/4 (in decimal form is: -0.75 is to the right of -1.25 on the number line)
    • At Least \geq
    • At Most \leq
    • Fewer \textless
    • More than \textgreater

 

  • Remember
    • 7 < 10 is not the same as 10 < 7
      • because 7 is less than 10, and 10 is greater than 7
      • NOT 10 is less than 7

 

  • Example 1: Solve for x
    • \frac{x}{-2}\textgreater9
    • Multiply -2 on both sides
      • -2(\frac{x}{-2})\textgreater(-2)9
    • Flip the sign (because you are multiplying by a negative)
      • x\textless-18

 

  • Example 2
    • Alyssa is saving up for a computer and knows she’ll need to spend at least $750 to get the one she wants. She saves $80 each month. How long will she need to save before she can go computer shopping?
      • The problem is asking for the variable “How long (in months)?”
      • You know that the side of the inequality with the variable will be at least (greater than or equal to) $750
        • Variable\geq750
      • You know she saves $80 each month
    • The word “each” means MULTIPLY, so $80 times x
      • x is the number of months she is saving
        • 80x\geq750
        • You say this: Eight x is greater than or equal to seven hundred fifty
        • This is the same thing as 750 is less than or equal to eighty x
          • 750\geq80x

 

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