COINS Lamar, Mario, and Nick put the money from their pockets on the table. Lamar had more money than Nick since $1.70 > $1.40. Will this still be true if each boy donates half their money to the school fundraiser? 1. Divide each side of the inequality 1.70 > 1.40 by 2. Write the resulting inequality and decide if it is true or false. 2. Who would have more if Mario and Lamar tripled their money by doing lawn work at home? Explain. The examples above demonstrate additional properties of inequality. These properties are also true for a ≥ b and a ≤ b. Solve Inequalities by Dividing 1 Solve 7y > -42. Check your solution. 7y > -42 Write the inequality. 7y _ 7 > -42 _ 7 Divide each side by 7. y > -6 Simplify. The solution is y > -6. Solving Inequalities by Multiplying or Dividing Checking Solutions Checking Solutions You can check the solution in Example 1 by substituting numbers greater than -6 into the inequality and testing it to verify that it holds true. 8-8 Name Amount of Money Lamar 1 dollar bill, 2 quarters, 2 dimes Mario 1 dollar bill, 3 quarters, 1 dime, 1 nickel Nick 5 quarters, 1 dime, 1 nickel MAIN IDEA Solve inequalities by using the Multiplication or Division Properties of Inequality. Math Online glencoe.com • Extra Examples • Personal Tutor • Self-Check Quiz Words When you multiply or divide each side of an inequality by a positive number, the inequality remains true. Symbols For all numbers a, b, and c, where c > 0, 1. if a > b, then ac > bc and a _ c > b _ c . 2. if a < b, then ac < bc and a _ c < b _ c . Examples 5 < 8 2 > -10 4(5) < 4(8) 2 _ 2 > -10 _ 2 20 < 32 1 > -5 Key Concept Properties of Inequality Lesson 8-8 Solving Inequalities by Multiplying or Dividing 449
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COINS Lamar, Mario, and Nick put the
money from their pockets on the table.
Lamar had more money than Nick
since $1.70 > $1.40. Will this still be
true if each boy donates half their
money to the school fundraiser?
1. Divide each side of the inequality
1.70 > 1.40 by 2. Write the resulting
inequality and decide if it is true or
false.
2. Who would have more if Mario and Lamar tripled their money by
doing lawn work at home? Explain.
The examples above demonstrate additional properties of inequality.
These properties are also true for a ≥ b and a ≤ b.
Solve Inequalities by Dividing
1 Solve 7y > -42. Check your solution.
7y > -42 Write the inequality.7y
_ 7 >
-42 _
7 Divide each side by 7.
y > -6 Simplify.
The solution is y > -6.
Solving Inequalities by Multiplying or Dividing
Checking SolutionsChecking SolutionsYou can check the solution in Example 1 by substituting numbers greater than -6 into the inequality and testing it to verify that it holds true.
8-8
Name Amount of Money
Lamar1 dollar bill, 2 quarters, 2 dimes
Mario1 dollar bill, 3 quarters, 1 dime, 1 nickel
Nick5 quarters, 1 dime, 1 nickel
MAIN IDEASolve inequalities by using the Multiplication or Division Properties of Inequality.
Math Online
glencoe.com• Extra Examples• Personal Tutor• Self-Check Quiz
Words When you multiply or divide each side of an inequality by a positive number, the inequality remains true.
Symbols For all numbers a, b, and c, where c > 0,
1. if a > b, then ac > bc and a _ c > b _ c .
2. if a < b, then ac < bc and a _ c < b _ c .
Examples 5 < 8 2 > -10
4(5) < 4(8) 2 _ 2 > -10 _ 2
20 < 32 1 > -5
Key ConceptProperties of Inequality
Lesson 8-8 Solving Inequalities by Multiplying or Dividing 449
The solution is x ≤ 24. You can check this solution by substituting 24
and a number less than 24 into the inequality.
Solve each inequality. Check your solution.
a. 3a ≥ 45 b. n _ 4 < -16 c. 81 ≤ 9p
What happens when each side of an inequality is multiplied or divided
by a negative number?
Graph 3 and 5 on a number line.
-2-3-4-5 543-1 210
Since 3 is to the left of 5, 3 < 5.
Multiply each number by -1.
-2-3-4-5 543-1 210
Since -3 is to the right of -5,
-3 > -5.
Notice that the numbers being compared switched positions as a result
of being multiplied by a negative number. In other words, their order
reversed.
These and other examples suggest the following properties. These
properties also hold true for a ≥ b and a ≤ b.
Common ErrorCommon ErrorDo not reverse the inequality symbol just because there is a negative sign in the inequality, as in 7y < -42. Only reverse the inequality symbol when you multiply or divide each side by a negative number.
Words When you multiply or divide each side of an inequality by a negative number, the direction of the inequality symbol must be reversed for the inequality to remain true.
Lesson 8-8 Solving Inequalities by Multiplying or Dividing 451
Multiply or Divide by a Negative Number
3 Solve a _ -2
≥ 8. Check your solution.
a _ -2
≥ 8 Write the inequality.
-2 ( a _ -2
) ≤ -2(8) Multiply each side by -2 and reverse the inequality symbol.
a ≤ -16 Check this result.
4 Solve -24 > -6n. Check your solution.
-24 > -6n Write the inequality.
-24 _ -6
< -6n _ -6
Divide each side by -6 and reverse the symbol.
4 < n or n > 4 Check this result.
d. c _ -7
< -14 e. -5d ≥ 30 f. -3 ≤ w _ -8
Some inequalities involve more than one operation. To solve, work backward as you did in solving two-step equations.
5 BASEBALL Manny was trying to break his school’s record by getting 61 hits in one season. Halfway through the season he already had 34 hits. Manny averages 2 hits per game. Write and solve an inequality to find how many more games it will take at that rate for Manny to have at least 61 hits. Interpret the solution.
The phrase at least means greater than or equal to. Let g = the number of games he needs to play. Then write an inequality.
34 + 2g ≥ 61 Write the inequality.
34 - 34 + 2g ≥ 61 - 34 Subtract 34 from each side.
2g ≥ 27 Simplify.
2g
_ 2 ≥ 27 _
2 Divide each side by 2.
g ≥ 13.5 Simplify.
If Manny plays only entire games, he should have 61 hits after 14 more games. Manny should break the record.
g. DVDS Joan has a total of $250. DVDs cost $18.95 each. Write and
solve an inequality to find how many DVDs she can buy and still have at least $50. Interpret the solution.