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An inequalitystates that two quantities

either are not equal or may not be

equal. An inequality uses one of the

following symbols:

is less than

Fewer than, below

is greater than

More than, above

is less than or equal to

At most, no more than

is greater than or equal to

At least, no less than

Graphing inequalities

- When a number is greater than or less than, you used an open dot on the graph.
- I think of the dot as being empty. It does not contain the number it is on.
- When a number is greater than or equal to or less than or equal to, you use a closed dot.
- I think of the dot as being full. It contains the number it is on.

Let’s try one!!!

3 + x < 9

WAIT!

The x is not isolated yet! What do we do??

That’s right! We do the inverse to isolate the variable. It’s the same way you have been solving the equations.

Let’s try it again!

- 3 + x < 9
- Isolate the variable by undoing the constant. Subtract 3 from both sides and you’re left with:

x < 6

- This would read x is less than 6. This means that x can be any number less than 6.
- In order to represent this, we put it on a graph.

Graphing the inequality

- x < 6
- Why did we use an open dot?
- Why did the arrow point to the left?

3 4 5 6 7 8 9 10

Let’s try a few more!

- X> 4.5
- X < -3
- X < -17
- X > 4

Lets try inequalities with words!

- All real numbers less than 5
- All real numbers greater than -1
- All real number less than or equal to ½
- All real number greater than or equal to 0

Write an inequality for the situation.

There are at most 87 people in the school.

Write an inequality for the situation.

At least 31 pencils are on the tray.

Additional Example 3: Money Application

Edgar’s August profit of $137 was at least $20 higher than his July profit. What was July’s profit?

Let p represent the profit increase from July to August.

August profitwas at least$20 higher thanJuly’s profit.

$137≥ 20 +p

137 ≥ 20 + p

Subtract 20 from both sides.

-20 -20

117≥ p

p ≤ 117

Rewrite the inequality.

July’s profit was at most $117.

Antonio has read at least 12 more books than Benji. If Benji has read 32 books, at least how many books has Antonio read?

2

– 6 > 1

y

2

> 7

y

2

> (2)7

(2)

Additional Example 1A: Solving Two-Step Inequalities

Solve. Then graph the solution set on a number line.

y

2

– 6 > 1

+ 6 + 6

Add 6 to both sides.

Multiply both sides by 2.

y > 14

Additional Example 1C: Solving Two-Step Inequalities

Solve.

4y – 5 < 11

4y – 5 < 11

+ 5 + 5

Add 5 to both sides.

4y < 16

4y < 16

Divide both sides by 4.

4

4

y < 4

252

, or 12

r ≤

Additional Example 2: Application

Sun-Li has $30 to spend at the carnival. Admission is $5, and each ride costs $2. What is the greatest number of rides she can ride?

Let r represent the number of rides Sun-Li can ride.

5 + 2r ≤ 30

– 5 –5

Subtract 5 from both sides.

2r ≤ 25

Divide both sides by 2.

2r ≤ 25

2 2

Sun-Li can ride only a whole number of rides, so the most she can ride is 12.

Brice has $30 to take his brother and his friends to the movies. If each ticket costs $4.00, and he must buy tickets for himself and his brother, what is the greatest number of friends he can invite?

Let t represent the number of tickets.

8 + 4t ≤ 30

– 8 –8

Subtract 8 from both sides.

4t≤ 22

Divide both sides by 4.

4t≤ 22

4 4

t ≤ 5.5

Brice can only buy a whole number of tickets, so the most people he can invite is 5.

Combining Like Terms

- Must have the same:
- Variable (Letter)
- Exponent (Power)
- Example:
- (2a + 3) + (a +7)

Steps for Solving a 2-Step Equation.

- Our Goal: Isolate the Variable!
- 1. Draw your slide to line up the equal signs.
- 2. Undo the constant by using the inverse operation.
- 3. Undo the variable term by using the inverse operation
- 4. Write the solution.

What if there are two variable terms?

- 2a = 7a +40

-7a -7a

-5a = 40

-5 -5

a = -8

- 1. Get the variable terms on one side, and constants on the other.
- Use inverse operation to isolate the variable.

Look for key words and translate them into mathematic symbols.

- Special cases
- More than: addition expression with the terms swapped
- Less than: subtraction expression with the terms swapped
- Divided into: division expression with the terms swapped

Luis’s cell phone plan costs $39 per month. Text messages cost an additional $0.15 each. If Luis’s cell phone bill last month totaled $55.05, write and solve an equation to determine how many text messages he sent.

Syreeta wants to buy some CDs that each cost $23. She has $65. Write and solve an equation to find how many CDs she can buy.

Toya had her birthday party at the movies. IT cost $27 for pizza and $8.50 per friend for the movie tickets. How many friends did Toya have at her party if she spent $78.

Joan bought a soft drink for 2 dollars and 7 candy bars. She spent a total of 16 dollars. How much did each candy bar cost?

Oceanside Bike Rental Shop charges 11 dollars plus 7 dollars an hour for renting a bike. Keith paid 53 dollars to rent a bike. How many hours did he pay to have the bike checked out?

Liam bought a movie ticket and a popcorn for $24. What was the price of each item if the ticket cost 3 times as much as the popcorn?

Doritos are on sale at Kroger. If Ms. Beaty buys 3 bags, it will cost her $6. If she buys them individually, she has to pay $3.99 each. How much money will she save on each bag if she buys 3.

Ms. Beaty can buy a Blu-Ray player for $45 and rent redbox movies for $10 a month or she can use her laptop and stream netflix for $15 per month. How many months would she need to stream netflix to pay as much as she would if she had bought the Ble-Ray and rented redbox movies?

One movie ticket at the Mall of Georgia costs $18, while one movie ticket at Regal costs $12. What is the difference in cost if you buy 5 tickets at each theater?

Jackie visited New York City and needed to take a cab from the airport to her hotel. The cab charges a $5.00 fee plus $0.20 per mile. If her total cost was $9.80, how many miles did Jackie ride in the cab?

Kristina rented a car for a road trip. She paid a rental fee of $60.00, as well as $9.75 each day. Her total cost was $138.00. For how many days did she rent the car?

Ms. Beaty found a Braves shirt that she wants to buy for her little brother’s birthday. The original price is $25. It is currently on sale at 15% off Ms. Beaty also has a $5 coupon. How much will the shirt cost before tax?

Yuan just bought a new television for $719.00. He made a down payment of $68.00 and will pay monthly payments of $31.00 until it is paid off. How many months will Yuan be paying? (Assume that Yuan pays no interest.)

A store is selling a couch for $873.00 on a 24-month payment plan. If a down payment of $121.00 is given, and no interest is charged, how much will a customer have to pay each month? Round your answer to the nearest cent.

Taylor needs to take a taxi to get to the movies. The taxi charges $4.00 for the first mile, and then $3.00 for each mile after that. If the total charge is $17.50, then how far was Taylor’s taxi ride to the movie?

Monday\'s low temperature in New York was 30°F. The same day in Dallas, the low temperature was above 57°F. at least how many degrees warmer was the low temperature in Dallas than the low in New York?

Erin spent $12.25 on supplies to make lemonade. At least how many glasses of lemonade must she sell at $0.30 per glass to make a profit?

Greg has $357.00. A DVD player costs $114.00, and he can purchase used movies for $8.85 each. What is the greatest number of movies Greg can buy if he also buys a DVD player?

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