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Solving Open Sentences Involving Absolute Value
5or2| xxx– 3 – 2 – 1 0 1 2 3 4 5 6| | | | | | | | | |
34| xx– 5 – 4 – 3 – 2 – 1 0 1 2 3 4| | | | | | | | | |
5or2| xxx
Solving Open Sentences Involving Absolute ValueThere are three types of open sentences that can involve absolute value.
nx nx nx 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
Case 1 The value inside the absolute value symbols is positive.
Case 2 The value inside the absolute value symbols is negative.
When solving equations that involve absolute value, there are two cases to consider:
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.
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
Answer: The solution set is
Write as or
Original inequality
Subtract 6 from each side.
Case 1 Case 2
Simplify.
Solve an Absolute Value Equation
Answer: {12, –2}
Solve 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
Check Substitute –4 and 6 into
Answer:
Write an Absolute Value Equation
Write an equation involving the absolute value for the graph.
Answer:
Write an Absolute Value Equation
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
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.
When solving equations of the form | x | < n, find the intersection of these two cases.
Then graph the solution set.
Write as and
Original inequality
Add 3 to each side.
Simplify.
Case 1 Case 2
Answer: The solution set is
Solve an Absolute Value Inequality (<)
Then graph the solution set.
Answer:
Solve an Absolute Value Inequality (<)
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
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.
When solving equations of the form | x | > n, find the union of these two cases.
Case 1 Case 2
Then graph the solution set.
Write as or
Add 3 to each side.
Simplify.
Original inequality
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:
Solve an Absolute Value Inequality (>)
Solving Open Sentences Involving Absolute Value
In general, there are three rules to remember when solving equations and inequalities involving absolute value:
1. If then or (solution set of two numbers)
2. If then and
(intersection of inequalities)
3. If then or(union of inequalities)
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