Text 6. Solve Linear Inequalities

ALGEBRA PROJECT UNIT 6 SOLVING LINEAR INEQUALITIES

SOLVING LINEAR INEQUALITIES

Lesson 1

Solving Inequalities by Addition and Subtraction

Lesson 2

Solving Inequalities by Multiplication and Division

Lesson 3

Solving Multi-Step Inequalities

Lesson 4

Solving Compound Inequalities

Lesson 5

Solving Open Sentences Involving Absolute Value

Lesson 6

Graphing Inequalities in Two Variables

SOLVING INEQUALITIES by ADDITION and SUBTRACTION

Example 1

Solve by Adding

Example 2

Graph the Solution

Example 3

Solve by Subtracting

Example 4

Variables on Both Sides

Example 5

Write and Solve an Inequality

Example 6

Write an Inequality to Solve a Problem

Solve

Then check your solution. Original inequality Add 12 to each side. This means all numbers greater than 77.

Check Substitute 77, a number less than 77, and a number greater than 77.

Answer: The solution is the set {all numbers greater than 77}.

Solve

Then check your solution.

Answer:

or {all numbers less than 14}

Solve

Then graph it on a number line. Original inequality Add 9 to each side. Simplify.

Answer: Since is the same as y ≤ 21, the solution set is

The heavy arrow pointing to the left shows that the inequality includes all the numbers less than 21.

The dot at 21 shows that 21 is included in the inequality.

Solve Answer:

Then graph it on a number line.

Solve

Then graph the solution. Original inequality Subtract 23 from each side. Simplify.

Answer: The solution set is

Solve Answer:

Then graph the solution.

Then graph the solution. Original inequality Subtract 12n from each side. Simplify. Answer: Since is the same as solution set is

the

Then graph the solution. Answer:

Write an inequality for the sentence below. Then solve the inequality. Seven times a number is greater than 6 times that number minus two. Seven times is greater six times a number than that number minus two.

7n

>

6n

–

Original inequality Subtract 6n from each side. Simplify. Answer: The solution set is

2

Write an inequality for the sentence below. Then solve the inequality. Three times a number is less than two times that number plus 5. Answer:

Entertainment Alicia wants to buy season passes to two theme parks. If one season pass cost $54.99, and Alicia has $100 to spend on passes, the second season pass must cost no more than what amount? Words

The total cost of the two passes must be less than or equal to $100.

Variable

Let

Inequality

the cost of the second pass. is less than The total cost or equal to

$100.

100

Solve the inequality. Original inequality Subtract 54.99 from each side. Simplify. Answer: The second pass must cost no more than $45.01.

Michael scored 30 points in the four rounds of the free throw contest. Randy scored 11 points in the first round, 6 points in the second round, and 8 in the third round. How many points must he score in the final round to surpass Michael’s score? Answer: 6 points

SOLVING INEQUALITIES by MULTIPLICATION and DIVISION

Example 1

Multiply by a Positive Number

Example 2

Multiply by a Negative Number

Example 3

Write and Solve an Inequality

Example 4

Divide by a Positive Number

Example 5

Divide by a Negative Number

Example 6

The Word “not”

Then check your solution. Original inequality Multiply each side by 12. Since we multiplied by a positive number, the inequality symbol stays the same. Simplify.

Check To check this solution, substitute 36, a number less that 36 and a number greater than 36 into the inequality.

Answer: The solution set is

Then check your solution. Answer:

Original inequality Multiply each side by change Simplify. Answer: The solution set is

and

Answer:

Write an inequality for the sentence below. Then solve the inequality. Four-fifths of a number is at most twenty. Four-fifths

of

×

a number

r

is at most

twenty.

20

Original inequality Multiple each side by

and do not

change the inequality’s direction. Simplify. Answer: The solution set is

.

Write an inequality for the sentence below. Then solve the inequality. Two-thirds of a number is less than 12. Answer:

Original inequality Divide each side by 12 and do not change the direction of the inequality sign. Simplify. Check

Answer: The solution set is

Answer:

using two methods. Method 1 Divide. Original inequality Divide each side by –8 and change < to >. Simplify.

Method 2 Multiply by the multiplicative inverse. Original inequality Multiply each side by and change < to >. Simplify. Answer: The solution set is

using two methods. Answer:

Multiple-Choice Test Item Which inequality does not have the solution A

B

C

D

Read the Test Item You want to find the inequality that does not have the solution set Solve the Test Item Consider each possible choice.

A.

B.

C.

D.

Answer: B

Multiple-Choice Test Item Which inequality does not have the solution A Answer: C

B

C

D

?

SOLVING MULT-STEP INEQUALITIES

Example 1

Solve a Real-World Problem

Example 2

Inequality Involving a Negative Coefficient

Example 3

Write and Solve an Inequality

Example 4

Distributive Property

Example 5

Empty Set

Science The inequality F > 212 represents the temperatures in degrees Fahrenheit for which water is a gas (steam). Similarly, the inequality represents the temperatures in degrees Celsius for which water is a gas. Find the temperature in degrees Celsius for which water is a gas.

Original inequality Subtract 32 from each side. Simplify. Multiply each side by Simplify. Answer: Water will be a gas for all temperatures greater than 100°C.

Science The boiling point of helium is –452°F. Solve the inequality to find the temperatures in degrees Celsius for which helium is a gas. Answer: Helium will be a gas for all temperatures greater than –268.9°C.

Then check your solution. Original inequality Subtract 13 from each side. Simplify. Divide each side by –11 and change Simplify.

Check To check the solution, substitute –6, a number than –6, and a number greater than –6.

Answer: The solution set is

less

Then check your solution. Answer:

Write an inequality for the sentence below. Then solve the inequality. Four times a number plus twelve is less than a number minus three. Four times a number 4n

is less than plus +

twelve 12

<

a number minus three.

Original inequality Subtract n from each side. Simplify. Subtract 12 from each side. Simplify. Divide each side by 3. Simplify. Answer: The solution set is

Write an inequality for the sentence below. Then solve the inequality. 6 times a number is greater than 4 times the number minus 2. Answer:

Original inequality Distributive Property Combine like terms. Add c to each side. Simplify. Subtract 6 from each side. Simplify. Divide each side by 4. Simplify.

Answer: Since

is the same as

the solution set is

Answer:

Original inequality Distributive Property Combine like terms. Subtract 4s from each side. This statement is false. Answer: Since the inequality results in a false statement, the solution set is the empty set Ø.

Answer: Ø

SOLVING COMPOUND INEQUALITIES

Example 1

Graph an Intersection

Example 2

Solve and Graph an Intersection

Example 3

Write and Graph a Compound Inequality

Example 4

Solve and Graph a Union

Graph the solution set of Graph

Graph

Find the intersection.

Answer: The solution set is Note that the graph of includes the point 5. The graph of does not include 12.

Graph the solution set of

and

Then graph the solution set. First express inequality.

using and. Then solve each and

The solution set is the intersection of the two graphs. Graph

Graph

Find the intersection.

Answer: The solution set is

Then graph the solution set. Answer:

Travel A ski resort has several types of hotel rooms and several types of cabins. The hotel rooms cost at most $89 per night and the cabins cost at least $109 per night. Write and graph a compound inequality that describes the amount that a quest would pay per night at the resort. Words

The hotel rooms cost at most $89 per night and the cabins cost at least $109 per night.

Variables Let c be the cost of staying at the resort per night. Inequality Cost per is at the is at night c

most $89 or

cost

89

c

or

least

$109. 109

Now graph the solution set. Graph

Graph

Find the union.

Answer:

Ticket Sales A professional hockey arena has seats available in the Lower Bowl level that cost at most $65 per seat. The arena also has seats available at the Club Level and above that cost at least $80 per seat. Write and graph a compound inequality that describes the amount a spectator would pay for a seat at the hockey game. Answer:

where c is the cost per seat

Then graph the solution set. or

Graph

Graph

Answer: Notice that the graph of contains every point in the graph of So, the union is the graph of The solution set is

Then graph the solution set. Answer:

SOLVING OPEN SENTENCES INVOLVING ABSOLUTE VALUE

Example 1

Solve an Absolute Value Equation

Example 2

Write an Absolute Value Equation

Example 3

Solve an Absolute Value Inequality (<)

Example 4

Solve an Absolute Value Inequality (>)

Method 1 Graphing means that the distance between b and –6 is 5 units. To find b on the number line, start at –6 and move 5 units in either direction.

The distance from –6 to –11 is 5 units. The distance from –6 to –1 is 5 units. Answer: The solution set is

Method 2 Compound Sentence Write

as

or

Case 1

Case 2 Original inequality Subtract 6 from each side. Simplify.

Answer: The solution set is

Answer: {12, –2}

Write an equation involving the absolute value for the graph.

Find the point that is the same distance from –4 as the distance from 6. The midpoint between –4 and 6 is 1.

The distance from 1 to –4 is 5 units. The distance from 1 to 6 is 5 units. So, an equation is .

Answer: Check Substitute –4 and 6 into

Write an equation involving the absolute value for the graph.

Answer:

Then graph the solution set. Write

as

and Case 2

Case 1 Original inequality Add 3 to each side. Simplify. Answer: The solution set is

Then graph the solution set.

Answer:

Then graph the solution set. Write

as

or Case 2

Case 1 Original inequality Add 3 to each side. Simplify. Divide each side by 3. Simplify.

Answer: The solution set is

Then graph the solution set.

Answer:

GRAPHING INEQUALITIES IN TWO VARIABLES

Example 1

Ordered Pairs that Satisfy an Inequality

Example 2

Graph an Inequality

Example 3

Write and Solve an Inequality

From the set {(3, 3), (0, 2), (2, 4), (1, 0)}, which ordered pairs are part of the solution set for Use a table to substitute the x and y values of each ordered pair into the inequality. x

y

True or False

3

3

true

0

2

false

2

4

true

1

0

false

Answer: The ordered pairs {(3, 3), (2, 4)} are part of the solution set of . In the graph, notice the location of the two ordered pairs that are solutions for in relation to the line.

From the set {(0, 2), (1, 3), (4, 17), (2, 1)}, which ordered pairs are part of the solution set for Answer: {(1, 3), (2, 1)}

Step 1 Solve for y in terms of x. Original inequality Add 4x to each side. Simplify. Divide each side by 2. Simplify.

Step 2 Graph Since does not include values when the boundary is not included in the solution set. The boundary should be drawn as a dashed line. Step 3 Select a point in one of the half-planes and test it. Let’s use (0, 0). Original inequality false

y = 2x + 3

Answer: Since the statement is false, the half-plane containing the origin is not part of the solution. Shade the other half-plane. y = 2x + 3

Answer: Since the statement is false, the half-plane containing the origin is not part of the solution. Shade the other half-plane. Check Test the point in the other half-plane, for example, (–3, 1). Original inequality

Since the statement is true, the half-plane containing (–3, 1) should be shaded. The graph of the solution is correct.

y = 2x + 3

Answer:

Journalism Lee Cooper writes and edits short articles for a local newspaper. It generally takes her an hour to write an article and about a half-hour to edit an article. If Lee works up to 8 hours a day, how many articles can she write and edit in one day? Step 1 Let x equal the number of articles Lee can write. Let y equal the number of articles that Lee can edit. Write an open sentence representing the situation. Number of articles she can write x

plus +

hour

times

number of articles she can edit y

is up to 8 hours. 8

Step 2 Solve for y in terms of x. Original inequality Subtract x from each side. Simplify. Multiply each side by 2. Simplify.

Step 3 Since the open sentence includes the equation, graph as a solid line. Test a point in one of the half-planes, for example, (0, 0). Shade the half-plane containing (0, 0) since is true.

Answer:

Step 4 Examine the situation. • Lee cannot work a negative number of hours. Therefore, the domain and range contain only nonnegative numbers. • Lee only wants to count articles that are completely written or completely edited. Thus, only points in the half-plane whose x- and ycoordinates are whole numbers are possible solutions. • One solution is (2, 3). This represents 2 written articles and 3 edited articles.

Food You offer to go to the local deli and pick up sandwiches for lunch. You have $30 to spend. Chicken sandwiches cost $3.00 each and tuna sandwiches are $1.50 each. How many sandwiches can you purchase for $30?

Answer:

The open sentence that represents this situation is where x is the number of chicken sandwiches, and y is the number of tuna sandwiches. One solution is (4, 10). This means that you could purchase 4 chicken sandwiches and 10 tuna sandwiches.

THIS IS THE END OF THE SESSION

BYE!