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Lesson 1-5 Solving Inequalities

Lesson 1-6Solving Compound and Absolute Value Inequalities

Example 1Solve an Inequality Using Addition or Subtraction

Example 2Solve an Inequality Using Multiplicationor Division

Example 3 Solve a Multi-Step Inequality

Example 4 Write an Inequality

Solve Graph the solution set on a number line.

Original inequality

Subtract 4y from each side.

Simplify.

Subtract 2 from each side.

Simplify.

Rewrite with y first.

Answer: Any real number greater than –5 is a solution of this inequality.

A circle means that this point is not included in the solution set.

Answer:

Solve Graph the solution set on a number line.

Solve Graph the solution set on a number line.

–40 p Simplify.

p –40 Rewrite with p first.

Original inequality

Divide each side by –0.3, reversing the inequality symbol.

Answer: The solution set is

A dot means that this point is included in the solution set.

Solve Graph the solution set on a number line.

Answer:

Solve Graph the solution set on a

number line.

Original inequality

Multiply each side by 2.

Add –x to each side.

Divide each side by –3, reversing the inequality symbol.

Answer: The solution set is and is graphed below.

Solve Graph the solution set on a

number line.

Answer:

Consumer Costs Alida has at most $10.50 to spend at a convenience store. She buys a bag of potato chips and a can of soda for $1.55. If gasoline at this store costs $1.35 per gallon, how many gallons of gasoline can Alida buy for her car, to the nearest tenth of a gallon?

Explore Let the number of gallons of gasoline that Alida buys.

Plan The total cost of the gasoline is 1.35g. The cost of the chips and soda plus the total cost of the gasoline must be less than or equal to $10.50. Write an inequality.

The cost ofchips & soda plus

the costof gasoline

is less thanor equal to $10.50.

1.55 + 1.35g 10.50

Divide each side by 1.35.

Solve Original inequality

Subtract 1.55 from each side.

Simplify.

Simplify.

Answer: Alida can buy up to 6.6 gallons of gasoline for her car.

Examine Since is actually greater than 6.6, Alida

will have enough money if she gets no more

than 6.6 gallons of gasoline.

Rental Costs Jeb wants to rent a car for his vacation. Value Cars rents cars for $25 per day plus $0.25 per mile. How far can he drive for one day if he wants to spend no more that $200 on car rental?

Answer: up to 700 miles

Example 1 Solve an “and” Compound Inequality

Example 2 Solve an “or” Compound Inequality

Example 3 Solve an Absolute Value Inequality (<)

Example 4 Solve an Absolute Value Inequality (>)

Example 5Solve a Multi-Step Absolute Value Inequality

Example 6 Write an Absolute Value Inequality

SolveGraph the solution set on a number line.Method 1 Write the compound inequality using the word

and. Then solve each inequality.and

Method 2 Solve both parts at the same time by adding 2 to each part. Then divide each part by 3.

Graph the solution set for each inequality and find their intersection.

y 4

SolveGraph the solution set on a number line.

Answer:

Solve each inequality separately.or

Answer: The solution set is

Solve orGraph the solution set on a number line.

Answer:

SolveGraph the solution set on a number line.

You can interpret to mean that the distance between d and 0 on a number line is less than 3 units. To make true, you must substitute numbers for d that are fewer than 3 units from 0.

All of the numbers between –3 and 3 are less than 3 units from 0.

Solve Graph the solution set on a number line.

Answer: The solution set is

Notice that the graph of is the same as the graph of d > –3 and d < 3.

Answer:

Solve Graph the solution set on a number line.

You can interpret to mean that the distance between d and 0 on a number line is greater than 3 units. To make true, you must substitute values for d that are greater than 3 units from 0.

All of the numbers not between –3 and 3 are greater than 3 units from 0.

Answer: The solution set is

Notice that the graph of is the same as the

graph of

Solve Graph the solution set on a number line.

Solve Graph the solution set on a number line.

Answer:

SolveGraph the solution set on a number line.

Solve each inequality.

or

is equivalent to

Answer: The solution set is .

SolveGraph the solution set on a number line.

Answer:

Housing According to a recent survey, the average monthly rent for a one-bedroom apartment in one city is $750. However, the actual rent for any given one-bedroom apartment might vary as much as $250 from the average.

Write an absolute value inequality to describe this situation.Let the actual monthly rent.

The rent for an apartmentcan differ from the average by as much as $250.

Answer: 250

Answer: The solution set isThe actual rent falls between $500 and $1000, inclusive.

Solve the inequality to find the range of monthly rent.Rewrite the absolute value inequality as a compound inequality. Then solve for r.

–rr

–r

Health The average birth weight of a newborn baby is 7 pounds. However, this weight can vary by as much as 4.5 pounds.

a. Write an absolute value inequality to describe this situation.

b. Solve the inequality to find the range of birth weights for newborn babies.

Answer:

Answer: The birth weight of anewborn baby will fall between 2.5 pounds and 11.5 pounds, inclusive.

Explore online information about the information introduced in this chapter.

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