solving and graphing inequalities chapter 6 review
TRANSCRIPT
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