Solving Open Sentences Involving Absolute Value
x | x  2 or x  5
x | 4  x  3
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–3 –2 –1
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–5 –4 –3 –2 –1
x | x  2 or x  5
Solving Open Sentences Involving Absolute Value
There are three types of open sentences that can
involve absolute value.
x n
x n
x n
Consider the case | x | = n.
| x | = 5 means the distance between 0 and x is 5 units
If | x | = 5, then x = – 5 or x = 5.
The solution set is {– 5, 5}.
Solving Open Sentences Involving Absolute Value
When solving equations that involve absolute value,
there are two cases to consider:
Case 1
The value inside the absolute value symbols
is positive.
Case 2
The value inside the absolute value symbols
is negative.
Equations involving absolute value can be solved by
graphing them on a number line or by writing them as
a compound sentence and solving it.
Solve an Absolute Value Equation
Method 1 Graphing
means that the distance between b and –6
is 5 units. To find b on the number line, start at –6 and
move 5 units in either direction.
The distance from –6 to –11 is 5 units.
The distance from –6 to –1 is 5 units.
Answer: The solution set is
Solve an Absolute Value Equation
Method 2 Compound Sentence
Write
as
or
Case 1
Case 2
Original inequality
Subtract 6 from
each side.
Simplify.
Answer: The solution set is
Solve an Absolute Value Equation
Answer: {12, –2}
Write an Absolute Value Equation
Write an equation involving the absolute value for
the graph.
Find the point that is the same distance from –4 as the
distance from 6. The midpoint between –4 and 6 is 1.
The distance from 1 to –4 is 5 units.
The distance from 1 to 6 is 5 units.
So, an equation is
.
Write an Absolute Value Equation
Answer:
Check Substitute –4 and 6 into
Write an Absolute Value Equation
Write an equation involving the absolute value for
the graph.
Answer:
Solving Open Sentences Involving Absolute Value
Consider the case | x | < n.
| x | < 5 means the distance between 0 and x is LESS
than 5 units
If | x | < 5, then x > – 5 and x < 5.
The solution set is {x| – 5 < x < 5}.
Solving Open Sentences Involving Absolute Value
When solving equations of the form | x | < n, find the
intersection of these two cases.
Case 1
The value inside the absolute value symbols
is less than the positive value of n.
Case 2
The value inside the absolute value symbols
is greater than negative value of n.
Solve an Absolute Value Inequality (<)
Then graph the solution set.
Write
as
and
Case 2
Case 1
Original inequality
Add 3 to each side.
Simplify.
Answer: The solution set is
Solve an Absolute Value Inequality (<)
Then graph the solution set.
Answer:
Solving Open Sentences Involving Absolute Value
Consider the case | x | > n.
| x | > 5 means the distance between 0 and x is
GREATER than 5 units
If | x | > 5, then x < – 5 or x > 5.
The solution set is {x| x < – 5 or x > 5}.
Solving Open Sentences Involving Absolute Value
When solving equations of the form | x | > n, find the
union of these two cases.
Case 1
The value inside the absolute value symbols
is greater than the positive value of n.
Case 2
The value inside the absolute value symbols
is less than negative value of n.
Solve an Absolute Value Inequality (>)
Then graph the solution set.
Write
as
or
Case 2
Case 1
Original inequality
Add 3 to each side.
Simplify.
Divide each side by 3.
Simplify.
Solve an Absolute Value Inequality (>)
Answer: The solution set is
Solve an Absolute Value Inequality (>)
Then graph the solution set.
Answer:
Solving Open Sentences Involving Absolute Value
In general, there are three rules to remember when
solving equations and inequalities involving absolute
value:
1. If x  n then x  n or x  n
(solution set of two numbers)
2. If x  n then x  n and x  n
(intersection of inequalities)
3. If x  n then x  n or
(union of inequalities)
x  n
 n  x  n