Transcript 1.7 Solving Absolute Value Equations & Inequalities

Name:
Date:
Period:
Topic: Solving Absolute Value Equations & Inequalities
Essential Question: What is the process needed to solve
absolute value equations and inequalities?
Warm-Up:
Describe the similarities and differences between
equations and inequalities.
Home-Learning #2 Review
Quiz #7:
Recall :
Absolute value | x | : is the distance
between x and 0. If | x | = 8, then
– 8 and 8 is a solution of the
equation ; or | x |  8, then any
number between 8 and 8 is a
solution of the inequality.
Absolute Value (of x)
•
•
•
•
Symbol lxl
The distance x is from 0 on the number line.
Always positive
Ex: l-3l=3
-4
-3
-2
-1
0
1
2
Recall:
You can solve some absolute-value
equations using mental math. For
instance, you learned that the equation
| x | 3 has two solutions: 3 and 3.
To solve absolute-value equations, you
can use the fact that the expression inside
the absolute value symbols can be either
positive or negative.
Solving an Absolute-Value Equation:
Solve | x  2 |  5
Solve | 2x  7 |  5  4
Answer ::
Solving an Absolute-Value Equation
Solve | x  2 |  5
The expression x  2 can be equal to 5 or 5.
x  2 IS POSITIVE
|x2|5
x  2  5
x7
x  2 IS NEGATIVE
|x2|5
x  2  5
x  3
The equation has two solutions: 7 and –3.
CHECK | 7  2 |  | 5 |  5
| 3  2 |  | 5 |  5
Answer ::
Solve | 2x  7 |  5  4
SOLUTION
Isolate the absolute value expression on one side of the equation.
2x  7 IS POSITIVE
2x
2x  77 IS
IS NEGATIVE
NEGATIVE
| 2x  7 |  5  4
|| 2x
2x  77 ||  55  44
| 2x  7 |  9
2x  7  +9
|| 2x
2x  77 ||  99
2x
2x
9
2x 
 77
7 
 9
9
2x  16
x8
2x
2x  2
2
TWO SOLUTIONS
x  1
x  1
Solve the following Absolute-Value Equation:
Practice:
1) Solve 6x-3 = 15
2) Solve 2x + 7 -3 = 8
Answer ::
1) Solve 6x-3 = 15
6x-3 = 15 or
6x = 18 or
x = 3 or
6x-3 = -15
6x = -12
x = -2
* Plug in answers to check your solutions!
Answer ::
2) 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.
***Important NOTE***
3 2x + 9 +12 = 10
- 12 - 12
3 2x + 9 = - 2
3
3
2x + 9 = - 2
3
What about this absolute value equation? 3x – 6 – 5 = – 7
Solving an Absolute Value Inequality:
● Step 1: Rewrite the inequality as a conjunction or a
disjunction.
● If you have a  or  you are working with a
conjunction or an ‘and’ statement.
Remember: “Less thand”
● If you have a  or  you are working with a
disjunction or an ‘or’ statement.
Remember: “Greator”
● Step 2: In the second equation you must negate the
right hand side and reverse the direction of the
inequality sign.
● Solve as a compound inequality.
Ex: “and” inequality
4 x  9  21
• Becomes an “and” problem
Positive
Negative
4x – 9 ≤ 21
+9 +9
4x ≤ 30
4
4
x ≤ 7.5
4x – 9 ≥ -21
+9 +9
4x ≥ -12
4
4
x ≥ -3
-3
7
8
This is an ‘or’
statement.
(Greator).
Ex: “or” inequality
|2x + 1| > 7
2x + 1 > 7
–1 -1
2x > 6
2 2
x>3
or
-4
In the 2nd
inequality, reverse
the inequality sign
and negate the
right side value.
2x + 1 < - 7
–1 -1
2x < - 8
2
2
3
Solving Absolute Value Inequalities:
Solve | x  4 | < 3 and graph the solution.
Solve | 2x  1 | 3  6 and graph the solution.
Answer ::
Solve | x  4 | < 3
x  4 IS POSITIVE
|x4|3
x  4 IS NEGATIVE
|x4|3
x  4  3
x  4  3
x7
x1
Reverse
inequality symbol.
The solution is all real numbers greater than 1 and less than 7.
This can be written as 1  x  7.
Answer ::
Solve | 2x  1 | 3  6 and graph the solution.
2x + 1 IS POSITIVE
2x + 1 IS NEGATIVE
| 2x  1 |  3  6
| 2x  1 | 3  6
| 2x  1 |  9
| 2x  1 |  9
2x  1  +9
2x  1  9
Reverse
inequality
symbol.
2x  10
2x  8
x4
x  5
The solution is all real numbers greater than or equal
to 4 or less than or equal to  5. This can be written as the
compound inequality x   5 or x  4.
6 5 4 3 2 1
0
1
2
3
4
5
6
Solve and graph the following
Absolute-Value Inequalities:
3) 3x  2  3  11
4) |x -5| < 3
Solve
Answer
& graph.
::
3)
3x  2  3  11
• Get absolute value by itself first.
3x  2  8
• Becomes an “or” problem
3x  2  8 or 3x  2  8
3x  10 or
3x  6
10
x
or x  2
3
-2
3
4
Answer ::
This is an ‘and’ statement.
(Less thand).
4) |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
Rewrite.
In the 2nd inequality, reverse the
inequality sign and negate the
right side value.
Solve each inequality.
Graph the solution.
2
8
Solve and Graph
5) 4m - 5 > 7 or 4m - 5 < - 9
6) 3 < x - 2 < 7
7) |y – 3| > 1
8) |p + 2| + 4 < 10
9) |3t - 2| + 6 = 2
Home-Learning #3:
• Page 211 - 212 (18, 26,36, 40, 64)