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## Solving and Graphing Inequalities

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### Solving and Graphing Inequalities

Chapter 6

Rules for graphing your answers

- If the letter is on the left then we can follow the direction of the arrow
- We must mark the numbers
- < or > we use an open circle
- < or > we use a closed circle

Our answers will look like this and we will graph the answers on a number line

- X < 2
- X > -2
- Z > 1
- 0 < d

Now you practice:

- T > 1
- X >-1
- N < 0
- 4 >y

Our problems will look like this:

- We will use our Chapter 3 rules to solve:
- Draw tracks
- Count variables
- If one, numbers jump tracks
- If 2 or more where are they
- Same side, family members
- Different sides, letters jump
- Add or subtract the numbers
- Is there a number on the letters?
- If its an integer divide each side
- If it’s a fraction, flip and multiply
- if the integer or fraction is negative I must turn the arrow around to the opposite direction
- Graph the solution

- X + 5 > 3

More examples:

- X + 4 < 7
- N + 6 > 2
- 5 > a + 5
- -2 > n – 4
- X – 5 > 2
- P – 1 < -4
- -3 < y - 2

Sec 6.2Solving Equations using Multi or Div

- We must be able to identify if there is an integer on the letter or a fraction on the letter.
- If the integer or the fraction is negative then we MUST change the direction of the arrow in our answer

Examples

- a/4 < 4
- 4x > 20
- k/4 < ½
- 18 < 2k
- 6 < t/5
- -21 < 3y

Examples: Negative numbers

- -1/2 y < 5
- -12m > 18
- -8x < 20
- -1/5 p > 1
- -2/3 x < -5
- -24 < 6t

Word Problems

- Kayla wants to buy some posters for her dorm room. Posters are on sale for $6 each. Write and solve an inequality to determine how many posters she can buy and spend no more than $25.
- Crandell plans to take figure skating lessons. He can rent skates for $5 per lesson. He can buy skates for $75. For what number of lessons is it cheaper for him to buy rather than rent skates?

QuizSections 6.1 and 6.2

- X – 4 < -5
- -11 > y + 4
- -1/2 x > -5
- -3x < -27
- -3/4 x < -1/4

Sec 6.3; Solving Multi-step Inequalities

- 2y – 5 < 7
- 5 – x > 4
- 3(x + 2) < 7
- -2(x + 1) < 2
- 2x – 3 > 4x – 1

Sec 6.3; Solving Multi-step Inequalities

- 5n – 21 < 8n
- -3z + 15 > 2z
- X + 3 > 2x – 4
- 4y – 3 < -y + 12

Word Problems

- You plan to make and sell candles. You pay $12 for instructions. The materials for each candle cost $0.50. You plan to sell each candle for $2. Let x be the numbers of candles you sell. How many candles must be sold to make at least $300 profit.

Sec 6.4; Solving compound inequalities

- How do you combine two thoughts in English class?
- What are they called?
- What words do we use?

How to Solve:

These are “AND” problems. These problems must be re-written into two problems.

- -2 < x + 2 < 4
- -1 < x + 3 < 7
- -6 < -3x < 12
- 0 < x – 4 < 12

Word Problems

- In the summer it took a Pony Express rider about 10 days to ride from St. Joseph, Missouri to Sacramento, California. In winter it took as many as 16 days. Write an inequality to describe the number of days that the trip might have taken.

Word Problems

- Frequency is used to describe the pitch of a sound. Frequencies are measured in hertz. Write an inequality for the following
- Sound of a human voice: 85 hertz to 1100 hertz
- Sound of a bats signal: 10,000 hertz to 120,000 hertz
- Sound heard by a dog: 15 hertz to 50,000 hertz
- Sound heard by a dolphin: 150 hertz to 150,000 hertz

Sec 6.5; Solving Compound Inequalities

- These are called “OR” problems
- These problems are already written and ready to be solved

Examples

- X-4<3 or 2x> 18
- 3x + 1 < 4 or 2x – 5 > 7
- X + 5 < -6 or 3x > 12
- 6x – 5 < 7 or 8x + 1 > 25

Word Problem

- A baseball is hit straight up in the air. Its initial velocity is 64 ft per second. Its formula is v = -32t + 64. Find the values of “t” for which the velocity of the baseball is greater than 32 or less than -32 feet per second.

Sec 6.6; Absolute Value Equations

- These problems will create some re-writing
- 1st – make sure the absolute value bars are on a side by themselves
- 2nd – drop the bars and write two problems
- 3rd – in the second problem you will need the opposite of the symbol (equals or inequality) and the opposite of the number.
- Now solve the equations
- If its an inequality then you will also graph the solutions.

Sec 6.7; Solving absolute value inequalities

- We will use the same rules as the previous section (6.6)
- Make sure to remember to make the changes to the second problem that you re-write.

Sec 6.8; Graphing Linear Inequalities in Two Variables

- We use our previous knowledge from Chapter’s 4 and 5.

To graph we will use y= mx + b

- Slope intercept form: y = mx + b
- “b” is the y-intercept. I must always use this number first when graphing. It is always located on the y-axis, either above or below the origin.
- “m” is the slope. Its always a fraction and remember to use rise over run

To graph given when given an equation; turn it into y=mx+b

- If the symbol is < or > we will draw a dashed line
- If the symbol is < or > we will draw a normal line
- To shade will have to use a test point and the slope intercept form
- If the test result is false shade away from the TP, if the test is true shade to the TP.

- Flow Chart
- Let x jump the tracks
- If there is a number on y, then we will set up three fractions and divide or reduce
- Collect “b” and find it on the y-axis
- Collect “m” and use rise over run to get to the next point on my line.

MORE EXAMPLES:

- X < -2
- Y < 1
- X + Y > 3
- 2X – Y > -2
- 3X – Y < 4

- P 371 #’S 36-50 EVEN

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