6 – 1 Solving Inequalities by Addition and Subtraction

• Method– Same as solving any equation– CHECK your answer

• Graphing– Greater than >

Less than < get an open circle

– Greater than or equal to Less than or equal to get an closed circle

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

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

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

Solve Then graph it on a number line.

Answer:

Solve Then graph the solution.

Original inequality

Subtract 23 from each side.

Simplify.

Answer: The solution set is

Solve Then graph the solution.

Answer:

Then graph the solution.

Original inequality

Subtract 12n from each side.Simplify.

Answer: Since is the same as the solution set is

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 timesa number

is greaterthan

six timesthat number minus two.

7n 6n 2> –

Simplify.Subtract 6n from each side.

Original inequality

Answer: The solution set is

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 the cost of the second pass.

Inequality 100

The total costis less thanor equal to $100.

Solve the inequality.

Answer: The second pass must cost no more than $45.01.

Original inequality

Subtract 54.99 from each side.

Simplify.

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

6 – 2 Solving Inequalities by Multiplication and Division

• Method– Same as solving any equation– One IMPORTANT exception

• When multiplying or dividing both sides of the inequality by a NEGATIVE NUMBER you MUST invert (change the direction of) the inequality sign

Then check your solution.

Original inequality

Multiply each side by 3. Since we multiplied by a positive number, the inequality symbol stays the same.

Simplify.

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

Answer: The solution set is

Then check your solution.

Answer:

Original inequality

Simplify.

Multiply each side by and change

Answer: The solution set is

Answer:

Write an inequality for the sentence below. Then solve the inequality.

Four-fifths of a number is at most twenty.

Four-fifths of is at most twenty.a number

r 20

Answer: The solution set is .

Original inequality

Simplify.

Multiple each side by and do not

change the inequality’s direction.

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.

Answer: The solution set is

Method 2 Multiply by the multiplicative inverse.

Original inequality

Multiply each side byand change < to >.

Simplify.

using two methods.

Answer:

Multiple-Choice Test Item

Which inequality does not have the solution

A B C D

Read the Test ItemYou want to find the inequality that does not have the solution set

Solve the Test Item

Consider each possible choice.

A.

D. C.

B.

Answer: B

Multiple-Choice Test Item

Which inequality does not have the solution ?

A B C D

Answer: C

6 – 3 Solving Multi-Step Inequalities

• Method– The same as solving any equation– Remember the negative multiplication/division

rule

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.

Answer: Water will be a gas for all temperaturesgreater than 100°C.

Original inequality

Subtract 32 from each side.

Simplify.

Multiply each side by

Simplify.

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 andchange

Simplify.

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

Answer: The solution set is

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 timesa number plus

is less than

a numberminus three. twelve

4n + <12

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

Add c to each side.Simplify.Subtract 6 from each side.Simplify.

Divide each side by 4.

Simplify.

Combine like terms.Distributive Property

Answer: Since is the same as

the solution set is

Answer:

Answer: Since the inequality results in a falsestatement, the solution set is the empty set Ø.

Original inequality

Distributive Property

Combine like terms.

Subtract 4s from each side.

This statement is false.

Answer: Ø

6 – 4 Solving Compound Inequalities

• Compound Inequality– An inequality containing two inequality

symbols– Types

• Intersection: only true if both parts of the inequality are true

• Union: true if any one part or all parts of the inequality are true

6 – 4 Solving Compound Inequalities

• Compound Inequality Graphing– Graphing Intersections

• Graph each part of the inequality and see where they overlap – that is the solution set

• x is greater than -2 AND x is less than or equal to 5 5

0 5-2

2 x

6 – 4 Solving Compound Inequalities

• Compound Inequality Graphing– Graphing Unions

• Graph each part of the inequality and see where they overlap – that is the solution set

12 9y or y

0 129

Graph the solution set of

Find the intersection.

Graph

Graph

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

Graph the solution set of and

Then graph the solution set.

First express using and. Then solve each inequality.

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 pernight

is atmost $89 or

thecost

is atleast $109.

c 89 109cor

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 ofThe solution set is

Then graph the solution set.

Answer:

6 – 5 Solving Open Sentences Involving Absolute Value

• Absolute Value– The distance from 0 on a number line

irrespective of direction.– Recall that means that x = 7 or x = -7

– Translations [IMPORTANT] • < and translate as an intersection (and)• > and translate as a union (or)

7x 7x

6 – 5 Solving Open Sentences Involving Absolute Value

• Solving – Recall solving a regular

absolute value equation– Now for a less than equation

3 5

3 5 3 5

2 8

x

x or x

x or x

7 12

7 12 7 12

5 19

x

x and x

x and x

Graph the answer and write an explanation as to why the graph is correct.

6 – 5 Solving Open Sentences Involving Absolute Value

• Solving a greater than equation

3 8

3 8 3 8

11 5

x

x or x

x or x

Graph the answer and write an explanation as to why the graph is correct.

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

Answer: The solution set is

Write as or

Original inequality

Subtract 6 from each side.

Case 1 Case 2

Simplify.

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 .

Check Substitute –4 and 6 into

Answer:

Write an equation involving the absolute value for the graph.

Answer:

Then graph the solution set.

Write as and

Original inequality

Add 3 to each side.

Simplify.

Case 1 Case 2

Answer: The solution set is

Then graph the solution set.

Answer:

Case 1 Case 2

Then graph the solution set.

Write as or

Add 3 to each side.

Simplify.

Original inequality

Divide each side by 3.

Simplify.

Answer: The solution set is

Then graph the solution set.

Answer:

6 – 6 Graphing Inequalities in Two Variables• Boundaries

– A line cuts the plane into 2 regions called half planes.– The line itself is a boundary

• Graphing, the Simple Method– Graph the inequalities as if they were a line – Then decide which region (s) contain the answers– Recall

• > means above the line but does not include the line

• < means below the line but does not include the line

• So and must include the line!

6 – 6 Graphing Inequalities in Two Variables

• Practice– Graph y > 3x + 2– Think y = 3x + 2

• Slope?• Y-intercept?

– Questions• Is the line included

or not?

x

y

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.

false01

true42

false20

true33

True or Falseyx

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 inequalityAdd 4x to each side.

Simplify.

Divide each side by 2.

Simplify.

Step 2 GraphSince 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 plus times

number of articles

she can edit is up to 8 hours.

hour

x + 8y

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 y- coordinates 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.

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