# 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
• $750\geq80x$