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Lesson 7-1 Solving Equations with Variables on Each Side
Lesson 7-2 Solving Equations with Grouping Symbols
Lesson 7-3 Inequalities
Lesson 7-4 Solving Inequalities by Adding or Subtracting
Lesson 7-5 Solving Inequalities by Multiplying or Dividing
Lesson 7-6 Solving Multi–Step Inequalities
Example 1 Equations with Variables on Each Side
Example 2 Equations with Variables on Each Side
Example 3 Use an Equation to Solve a Problem
Solve . Check your solution.
Write the equation.
Subtract 5x from each side.
Simplify.
Mentally divide each side by –3Subtract 5x from the left side of the equation to isolate the variable.
Subtract 5x from the right side of the equation to keep it balanced.
Check
This statement is true.
Write the equation.
Answer: The solution is –4.
Check to see whether this statement is true.
Replace x with –4.
To check your solution, replace x with –4 in the original equation.
Solve . Check your solution.
Answer: 3
Solve .
Answer: The solution is 4.
Write the equation.
Subtract 2x from each side.
Simplify.
Subtract 3 from each side.
Simplify.
Mentally divide each side by 5.
Answer: The solution is 4.4.
Solve .
Write the equation.
Subtract a from each side.
Simplify.
Add 4.9 to each side.
Simplify.
Divide each side by 1.5.
Check your solution.
a. Solve .
b. Solve .
Answer: –11
Answer: 1.2
Car Rental A car rental agency has two plans. Under plan A, a car rents for $80 plus $20 each day. Under plan B, a car rents for $120 plus $15 each day.What number of days results in the same cost?
Let d represent the number of days.
Words $80 plus $20 for each day $120 plus $15 for each day
Variables
Answer: The cost would be the same for 8 days.
Equation
Write the equation.
Subtract 15d from each side.
Simplify.
Subtract 80 from each side.
Simplify.
Divide each side by 5.
Simplify.
Cell Phones A cell phone provider offers two plans. Under plan A, the monthly cost is $20 with a cost of $0.35 per minute. Under plan B, the monthly cost is $35 with a cost of $0.15 per minute. What number of minutes results in the same cost?
Answer: 75 minutes
Example 1 Solve Equations with Parentheses
Example 2 Use an Equation to Solve a Problem
Example 3 No Solution
Example 4 All Numbers as Solutions
Solve . Check your solution.
Write the equation.
Use the Distributive Property.
Simplify.
Subtract 5h from each side.
Simplify.
Divide each side by –2.
Simplify.
Check
Answer: The solution is 5.
Write the equation.
This statement is true.
Replace h with 5.
Simplify.
Answer: The solution is 12.5.
Solve .
Write the equation.
Use the Distributive Property.
Subtract 3b from each side.
Simplify.
Add 12 to each side.
Simplify.
Divide each side by 3.
Simplify.
a. Solve . Check your solution.
b. Solve .
Answer: 7
Answer: 9
Geometry The perimeter of a rectangle is 36 inches. Find the dimensions if the length is 2 inches greater than three times the width.
Words The length is 2 inches greater than three times the width. The perimeter is 36 inches.
Variables Let the width.
Let the length.
2 times the length 2 times the width perimeter
Equation
Evaluate to find the length.
Solve .
Use the Distributive Property.
Simplify.
Subtract 4 from each side.
Simplify.
Mentally divide each side by 8.
Write the equation.
Replace w with 4.
Check Add the lengths of the four sides.
Answer: The width is 4 inches. The length is 14 inches.
Geometry The perimeter of a rectangle is 26 feet. Find the dimensions if the length is 2 feet less than twice the width.
Answer: width = 5 feet; length = 8 feet
Solve .
Subtract 4x from each side.
Simplify.
Write the equation.
Answer: The sentence is never true.
So, the solution set is .
Answer: The solution set is .
Solve .
Solve .
Write the equation.
Use the Distributive Property.
Subtract 9 from each side.
Simplify.
Simplify.
Mentally divide each side by 12.
Answer: The sentence is always true.The solution set is all numbers.
Answer: The solution set is all numbers.
Solve .
Example 1 Write Inequalities with < or >
Example 2 Write Inequalities with or
Example 3 Use an Inequality
Example 4 Determine Truth of an Inequality
Example 5 Graph Inequalities
Example 6 Write an Inequality
Write an inequality for the following sentence.
Your age is less than 19 years.
Variable Let a represent age.
Answer:
Inequality
Write an inequality for the following sentence.
Your height is greater than 52 inches.
Variable Let h represent height.
Answer:
Inequality
Write an inequality for each sentence.
a. Your height is less than 48 inches.
b. Your age is greater than 12 years.
Answer:
Answer:
Write an inequality for the following sentence.
Your speed is less than or equal to 62.
Variable Let s represent speed.
Inequality
Answer:
Answer:
Write an inequality for the following sentence.
Your speed is greater than or equal to 42.
Variable Let s represent speed.
Inequality
Write an inequality for each sentence.
a. Your weight is less than or equal to 120 pounds.
b. Your speed is greater than or equal to 35.
Answer:
Answer:
Air Pollution To meet a certain air quality standard, an automobile must have a fuel efficiency of not less than 27.5 miles per gallon. Write an inequality to describe this situation.
Words Fuel efficiency of not less than 27.5 miles per gallon
Variables Let fuel efficiency in miles per gallon.
Answer: The inequality is .
Inequality
Cafeteria The school cafeteria allows each student no more than 2 servings of dessert during lunch. Write an inequality to describe this situation.
Answer:
Answer: The sentence is true.
For the given value, state whether the inequality is true or false.
Write the inequality.
Simplify.
Replace s with 6.
Answer: The sentence is false.
For the given value, state whether the inequality is true or false.
Write the inequality.
Replace a with 36.
Simplify.
Simplify.
For the given value, state whether each inequality is true or false.
a.
b.
Answer: false
Answer: true
Graph .
The open circle means the number 10 is not included in the graph.
Answer:
The closed circle means the number 10 is included in the graph.
Graph .
Answer:
The open circle means the number 10 is not included in the graph.
Graph .
Answer:
The closed circle means the number 10 is included in the graph.
Graph .
Answer:
Graph each inequality.
a.
b.
c.
d.
Answer:
Answer:
Answer:
Answer:
Write the inequality for the graph.
A closed circle is on –38, so the point –38 is included in the graph. The arrow points to the right, so the graph includes all numbers greater than or equal to –38.
Answer: The inequality is .
Write the inequality for the graph.
Answer: The inequality is .
Example 1 Solve an Inequality Using Subtraction
Example 2 Solve an Inequality Using Addition
Example 3 Graph Solutions of Inequalities
Example 4 Use an Inequality to Solve a Problem
Solve . Check your solution.
Subtract 5 from each side.
Simplify.
To check your solution, try any number greater than 6.
Write the inequality.
This statement is true.
Answer: The solution is .
Write the inequality.Check
Replace y with 7.
Solve .
Answer: The solution is .
Solve . Check your solution.
Answer: The solution is .
Add 8 to each side.
Simplify.
Check You can check your result by replacing d in theoriginal inequality with a number less than orequal to –13.
Write the inequality.
Solve . Check your solution.
Answer:
Solve . Graph the solution on a number line.
Add to each side.
Write the inequality.
Answer: The solution is .
Simplify.
Graph the solution.
Answer:
Place an open circle at . Draw a line and arrow to the left.
Solve . Graph the solution on a number line.
Answer:
Bowling Katya has $12 to take to the bowling alley. If the shoe rental costs $3.75, what is the most she can spend on games and snacks?
Explore We need to find the greatest amount Katya can spend on games and snacks.
Plan Let x represent the amount Katya can spend on games and snacks. Write an inequality to represent the problem. Recall that at most means less than or equal to.
Cost ofshoes
cost of gamesand snacks
must be less than or equal toplus total
Subtract 3.75 from each side.
Simplify.
Solve Write the inequality.
Examine Check by choosing an amount less than or equal to $8.25, say, $6. Then Katya would spend $3.75 + $6 or $9.75 in all. Since $9.75 < $12, the answer is reasonable.
Answer: The most Katya can spend on games and snacks is $8.25.
Movies Danielle has $10 to take to the movies. If the cost of a ticket is $4.50, what is the most she can spend on snacks?
Answer: $5.50
Example 1 Multiply or Divide by a Positive Number
Example 2 Write an Inequality
Example 3 Multiply or Divide by a Negative Number
Solve . Check your solution.
Divide each side by 9.
Simplify.
Write the inequality.
Answer: The solution is . You can check this solution by substituting 6 or a number less than 6 into the inequality.
Answer: The solution is . You can check this solution by substituting a number greater than 36 into the inequality.
Solve . Check your solution.
Write the inequality.
Multiply each side by 9.
Simplify.
a. Solve . Check your solution.
b. Solve . Check your solution.
Answer:
Answer:
Multiple-Choice Test Item
Martha earns $9 per hour working for a fast-food restaurant. Which inequality can be used to find how many hours she must work in a week to earn at least $243?
A B C D
Read the Test ItemYou are to write an inequality to represent a real-world problem.
Solve the Test ItemLet x represent the number of hours worked.
Amount earned per hour times
number of hours
is at least
amount earnedeach week.
Answer: The answer is B.
Multiple-Choice Test Item
Ed earns $6 per hour working at the library. Which inequality can be used to find how many hours he must work in a week to earn more than $100?
A B C D
Answer: D
Solve and check your solution. Then graph the solution on a number line.
Check You can check your result by replacing x in the original inequality with a number less than –35.
Answer: Check this result.
Multiply each side by –5 and reverse the symbol.
Write the inequality.
Answer:
Graph the solution, .
Solve and check your solution. Then graph the solution on a number line.
Write the inequality.
Answer: Check this result.
Divide each side by –9 and reverse the symbol.
Graph the solution, .
Answer:
Solve each inequality and check your solution. Then graph the solution on a number line.
a.
b.
Answer:
Answer:
Example 1 Solve a Two–Step Inequality
Example 2 Reverse the Inequality Symbol
Example 3 Inequalities with Grouping Symbols
Solve and check your solution. Graph the solution on a number line.
Write the inequality.
Subtract 13 from each side.
Simplify.
Divide each side by 5.
Answer: Simplify.
To check your solution, try 15, a number greater than 14.
The solution checks.
Check Write the inequality.
Simplify.
Graph the solution, .
Answer:
Replace x with 15.
Solve and check your solution. Graph the solution on a number line.
Answer:
Solve and check your solution. Graph the solution on a number line.
Write the inequality.
Add 2a to each side.
Simplify.
Subtract 7 from each side.
Simplify.
Divide each side by –2 and change to .
Simplify.Answer:
Check
Try –7, a number greater than –8.
The solution checks.
Graph the solution, .
Replace a with –7.
Simplify.
Solve and check your solution. Graph the solution on a number line.
Answer:
Backpacking A person weighing 213 pounds has a 10-pound backpack. If three times the weight of your backpack and its contents should be less than your body weight, what is the maximum weight for the contents of the pack?
Let c represent the weight of the contents of the pack.
Words 3 timesweight of pack and contents
should be less than body weight.
Inequality 3
Solve the inequality.
Answer: The weight of the contents should be less than 61 pounds.
Write the inequality.
Use the Distributive Property.
Subtract 30 from each side.
Simplify.
Divide each side by 3.
Simplify.
Backpacking A person weighing 168 pounds has a 7-pound backpack. If three times the weight of your backpack and its contents should be less than your body weight, what is the maximum weight for the contents of the pack?
Answer: Less than 49 pounds
