Solving and Graphing

Inequalities

CHAPTER 6 REVIEW

An inequality is like an equation, but instead of an equal sign (=) it

has one of these signs:

< : less than≤ : less than or equal to

> : greater than≥ : greater than or equal to

“x < 5”

means that whatever value x has, it must be less than 5.

Try to name ten numbers that are less than 5!

Numbers less than 5 are to the left of 5 on the number line.

0 5 10 15-20 -15 -10 -5-25 20 25

• If you said 4, 3, 2, 1, 0, -1, -2, -3, etc., you are right.• There are also numbers in between the integers, like 2.5, 1/2, -7.9, etc. • The number 5 would not be a correct answer, though, because 5 is not less than 5.

“x ≥ -2”means that whatever value x has, it must be greater than or

equal to -2.

Try to name ten numbers that are greater than or equal to -2!

Numbers greater than -2 are to the right of 5 on the number line.

0 5 10 15-20 -15 -10 -5-25 20 25

• If you said -1, 0, 1, 2, 3, 4, 5, etc., you are right.• There are also numbers in between the integers, like -1/2, 0.2, 3.1, 5.5, etc. • The number -2 would also be a correct answer, because of the phrase, “or equal to”.

-2

Where is -1.5 on the number line? Is it greater or less than -2?

0 5 10 15-20 -15 -10 -5-25 20 25

• -1.5 is between -1 and -2.• -1 is to the right of -2.• So -1.5 is also to the right of -2.

-2

Solve an Inequality

w + 5 < 8

w + 5 + (-5) < 8 + (-5)

w < 3All numbers less

than 3 are solutions to this

problem!

More Examples

8 + r ≥ -2

8 + r + (-8) ≥ -2 + (-8)

r ≥ -10All numbers from -10 and up (including

-10) make this problem true!

More Examples

x - 2 > -2

x + (-2) + (2) > -2 + (2)

x > 0

All numbers greater than 0 make this problem true!

More Examples

4 + y ≤ 1

4 + y + (-4) ≤ 1 + (-4)

y ≤ -3

All numbers from -3 down (including -3) make this problem true!

There is one special case.

●Sometimes you may have to reverse the direction of the inequality sign!!

●That only happens when you

multiply or divide both sides of the inequality by a negative number.

Example:

Solve: -3y + 5 >23

-5 -5

-3y > 18

-3 -3

y < -6

●Subtract 5 from each side.

●Divide each side by negative 3.

●Reverse the inequality sign.

●Graph the solution.

0-6

Try these:1.) Solve 2x + 3 > x + 5 2.)Solve - c – 11 >23

3.) Solve 3(r - 2) < 2r + 4

-x -x

x + 3 > 5 -3 -3

x > 2

+ 11 + 11

-c > 34 -1 -1

c < -34

3r – 6 < 2r + 4-2r -2r

r – 6 < 4 +6 +6 r < 10

You did remember to reverse the signs . . .

57415 x771248 x

4

7

4 42 x 3

. . . didn’t you?Good job!

Example: 8462 xx- 4x - 4x

862 x + 6 +6

142 x -2 -2

Ring the alarm! We divided by a

negative!

7xWe turned the sign!

Remember Absolute Value

Ex: Solve 6x-3 = 15

6x-3 = 15 or 6x-3 = -15

6x = 18 or 6x = -12

x = 3 or x = -2

* Plug in answers to check your solutions!

Ex: Solve 2x + 7 -3 = 8

Get the abs. value part by itself first!

2x+7 = 11

Now split into 2 parts.

2x+7 = 11 or 2x+7 = -11

2x = 4 or 2x = -18

x = 2 or x = -9

Check the solutions.

Ex: Solve & graph.

• Becomes an “and” problem

2194 x

2

153 x

-3 7 8

Solve & graph.

• Get absolute value by itself first.

• Becomes an “or” problem

11323 x

823 x

823or 823 xx63or 103 xx

2or 3

10 xx

-2 3 4

Example 1:● |2x + 1| > 7

● 2x + 1 > 7 or 2x + 1 >7

● 2x + 1 >7 or 2x + 1 <-7

● x > 3 or x < -4

This is an ‘or’ statement. (Greator). Rewrite.

In the 2nd inequality, reverse the inequality sign and negate the right side value.

Solve each inequality.

Graph the solution.

3-4

Example 2:● |x -5|< 3

● x -5< 3 and x -5< 3

● x -5< 3 and x -5> -3

● x < 8 and x > 2

● 2 < x < 8

This is an ‘and’ statement. (Less thand).

Rewrite.

In the 2nd inequality, reverse the inequality sign and negate the right side value.

Solve each inequality.

Graph the solution.

8 2

Absolute Value InequalitiesCase 1 Example:

3 5x

3 5

2

x

x

and

8x

53x

8x2

Absolute Value InequalitiesCase 2 Example:

2 1 9x

2 1 9

2 10

5

x

x

x

5x

2 1 9

2 8

4

x

x

x

4x OR

or

Absolute Value• Answer is always positive

• Therefore the following examples

cannot happen. . .

Solutions: No solution

95 3x

Graphing Linear Inequalities in Two

Variables•SWBAT graph a linear

inequality in two variables

•SWBAT Model a real life situation with a linear

inequality.

Some Helpful Hints

•If the sign is > or < the line is dashed

•If the sign is or the line will be solid

When dealing with just x and y.

•If the sign > or the shading either goes up or to the right

•If the sign is < or the shading either goes down or to the left

When dealing with slanted lines

•If it is > or then you shade above

•If it is < or then you shade below the line

Graphing an Inequality in Two Variables

Graph x < 2Step 1: Start by graphing the line x = 2

Now what points would give you less

than 2?

Since it has to be x < 2 we shade everything to

the left of the line.

Graphing a Linear Inequality

Sketch a graph of y 3

Using What We KnowSketch a graph of x + y < 3

Step 1: Put into slope intercept form

y <-x + 3

Step 2: Graph the line y = -x + 3