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7/27/2019 1.7_Linear and Absolute Value Inequalities
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College Algebra: Section 1.7
Equations and InequalitiesInvolving Absolute Value
Objectives of this Section
Solve Equations Involving Absolute Value
Solve Inequalities Involving Absolute Value
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Equations Involving Absolute Value
If the absolute value of an expression equals
some positive numbera, then the expression
itself equals eithera or -a. Thus,u a u a u a is equivalent to or
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Solve: 2 3 11x
2 3 11x
2 3 11 2 3 11x x or
2 14x x 7
2 8x x 4
Solution set: {-4, 7}
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u a a u a
u a a u a
is equivalent to
is equivalent to
Theorem
In other words, |u| < a is equivalent
to -a < u and u < a.
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Ex. Solve y + 21 7
- 21 -21y -14
(-14) + 21 7
7 7
1. Draw the river
2. Subtract 21 fromboth sides
3. Simplify4. Check your answer
5. Graph the solution
-14 -13-15
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Ex. Solve 8y + 3 > 9y - 14
o17 1816
- 8y - 8y3 > y - 14
+ 14 + 1417 > y
y
< 178(16) + 3 > 9(16)14131 > 130
1. Draw the river2. Subtract 8y from both
sides
3. Simplify
4. Add 14 to both sides5. Simplify
6. Rewrite inequality
with the variable first
7. Check your answer8. Graph the solution
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Ex. Solve 8y + 3 > 9y - 14
o17 1816
- 8y - 8y3 > y - 14
+ 14 + 1417 > y
y
< 178(16) + 3 = 9(16)14131 > 130
Big Tip!!!
At the end of solving
inequality, always put the
variable at the LEFT hand
side. Then arrow of the
inequality sign tells you the
correct graph
y < 17
The graph should toward to
the left.
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To Solve the Absolute Value Inequalities
1. Isolate the absolute value expression.
2. Make sure the absolute value inequality can be defined.
3. For any defined absolute value inequality with template:|X| a, where a> 0 and = , ,
write the corresponding compound inequalities by following the
rules:
a) if |X| > a or |X| a, meaning greator, leaving
the jail
b) set jail boundaries as a and ac) write compound inequalities as
X a meaning stay left to the left
boundary or right to the right boundary
d) if |X| < a or |X| a, meaning less thand, going
to the jail
e) set jail boundaries as a and a
f) write compound inequalities as
a< X < a (and) meaning stay between the two
boundaries
4. Solve the converted compound inequalities.
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Solve | x3 | < 5
Absolute Value Inequality
(1) The absolute value expression is isolated(2) It is a well defined absolute value inequality
(3) It is a less thand inequality. (go to the jail)
a) set jail boundaries: 5, and 5
b) write compound inequalities: (stay in between the jailboundaries)
5 < x3 < 5
2 < x < 8
+3 +3 +3
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You try this!
Solve | x + 4 | < 1
Absolute Value Inequality
(1) The absolute value expression is isolated
(2) It is a well defined absolute value inequality
(3) It is a less thand inequality. (go to the jail)
a) set jail boundaries: 1, and 1
b) write compound inequalities: (stay in between the jail
boundaries)
1 < x + 4 < 1
5 < x
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Solve | 2x + 3 |3 2
Absolute Value Inequality
(1) The absolute value expression is NOT isolated
| 2x + 3 |3 2
| 2x+ 3 | 5
(2) It is a well defined absolute value inequality(3) It is a greator inequality. (leaving the jail)
a) set jail boundaries: 5, and 5
b) write compound inequalities: (stay left to left boundary
and right to the right boundary)
2x + 3 5 or 2x + 3 5
2x 8 or 2x 2
x 4 or x 1
3 3
+ 3 + 3
3 3
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You try this!
Solve | 4x3 | + 5 8
Absolute Value Inequality
(1) The absolute value expression is NOT isolated
| 4x3| + 5 8
| 4x3 3
(2) It is a well defined absolute value inequality(3) It is a greator inequality. (leaving the jail)
a) set jail boundaries: 3, and 3
b) write compound inequalities: (stay left to left boundary
and right to the right boundary)
4x3 3 or 4x3 3
4x 0 or 4x 6
x 0 or x 3/2
+ 3 + 3
5 5
+ 3 + 3
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Summary
1. Before solving absolute value equations or inequalities, you
MUST isolate the absolute value expression and check themare defined or not.
2. Remember telling yourself the jail story. It will help you set
up the correct equations or compound inequalities.
3. Solve the equations or compound inequalities and dont
forget the learned knowledge such as when multiplying or
dividing a negative number, you need flip the inequality
sign.
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Solve: 3 1 5x
3 1 5x
5 3 1 5x
4 3 6x
4
32x
Solution Set:
2,3
42
3
4| orxx
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u a u a u a
u a u a u a
is equivalent to or
is equivalent to or
Ifa is any positive number, then
Theorem
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Solve: 4 3 15x
4 3 15x
4 3 15x or 4 3 15x
4 18x
x 9
2
4 12x
x 3
),3[2
9,3
2
9|
orxorxx