Transcript 1-5 Absolute Value Equations & Inequalities

1-5 Absolute Value
Equations & Inequalities
M11.D.2.1.1: Solve compound inequalities and/or
graph their solution sets on a number line.
Vocabulary
The absolute value of a number is its distance from
zero on the number line and distance is
nonnegative.
Key Concept
Algebraic Definition of Absolute Value
If x ≥ 0, then |x| = x
If x < 0, then |x| = -x
Definition Example
Let’s look at
|z + 2| = 10
Since z+2 is inside the absolute value bracket, the
solution can either be positive or negative.
So, z + 2 = 10 or z + 2 = -10
Solving Absolute Value
Equations
Solve |15 – 3x| = 6.
|15 – 3x| = 6
15 – 3x = 6
or
15 – 3x = –6
–3x = –9
x =3
Check:
or
|15 – 3x| = 6
|15 – 3(3)|
6
|6|
6
6 = 6
The value of 15 – 3x can be 6 or –
6 since |6| and |–6| both equal 6.
–3x = –21
Subtract 15 from each side of both
equations.
x=7
Divide each side of both equations
by –3.
|15 – 3x| = 6
|15 – 3(7)|
6
|–6|
6
6 = 6
Solving Multi-Step Absolute
Value Equations
Solve 4 – 2|x + 9| = –5.
4 – 2|x + 9| = –5
–2|x + 9| = –9
Add –4 to each side.
|x + 9| = 92
x + 9 = 92 or x + 9 = –92
x = –4.5 or
Check:
x = –13.5
4 – 2 |x + 9| = –5
4 – 2 |–4.5 + 9|
–5
4 – 2 |4.5|
–5
4 – 2 (4.5)
–5
–5 = –5
Divide each side by –2.
Rewrite as two equations.
Subtract 9 from each side
of both equations.
4 – 2|x + 9| = –5
4 – 2 |–13.5 + 9|
–5
4 – 2 |–4.5|
–5
4 – 2 (4.5)
–5
–5 = –5
Vocabulary
An extraneous solution is a solution of an equation
derived from an original equation that is not a solution
of the original equation.
Checking for Extraneous
Solutions
Solve |3x – 4| = –4x – 1.
|3x – 4| = –4x – 1
3x – 4 = –4x – 1
or 3x – 4 = –(–4x –1)
7x – 4 = –1
3x – 4 = 4x + 1
– x=5
7x = 3
x = 3
7
or
x = –5
Rewrite as two
equations.
Solve each equation.
Continued…
(continued)
Check: |3x – 4| = –4x – 1
3
|3( 7 ) – 4|
|– 19 |
7
3
–4( 7 ) – 1
– 19
|3x – 4| = –4x – 1
|3(–5) – 4|
|–19|
7
19
19
=
/
7
7
(–4(–5) –1)
19
19 = 19
The only solution is –5. 3 is an extraneous solution.
7
Key Concepts
Let k represent a positive real number.
|x| ≥ k is equivalent to
x ≤ -k or x ≥ k
|x| ≤ k is equivalent to
-k ≤ x ≤ k
Solving Absolute Value
Inequalities, |A| ≥ b
Solve |2x – 5| > 3. Graph the solution.
|2x – 5| > 3
2x – 5 < –3 or 2x – 5 > 3 Rewrite as a compound inequality.
2x < 2
x < 1
2x > 8 Solve for x.
or
x > 4
Solving Absolute Value
Inequalities, |A| < b
Solve –2|x + 1| + 5 >
– –3. Graph the solution.
–2|x + 1| + 5 >
– –3
–2|x + 1| >
– –8
|x + 1| <
–4
Isolate the absolute value expression.
Subtract 5 from each side.
Divide each side by –2 and reverse
the inequality.
–4 <
– x + 1<
–4
Rewrite as a compound inequality.
–5 <
–x<
–3
Solve for x.
Real-World Connection
The area A in square inches of a square photo is required to
satisfy 8.5 <
– A<
– 8.9. Write this requirement as an absolute value
inequality.
8.9 – 8.5
0.4
=
2
2 = 0.2
8.9 + 8.5
17.4
=
2
2 = 8.7
Find the tolerance.
Find the average of the maximum and
minimum values.
–0.2 <
– A – 8.7 <
– 0.2
Write an inequality.
|A – 8.7| <
– 0.2
Rewrite as an absolute value inequality.