Solving Absolute Value Equations and Inequalities

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Section 2.8

• Solving Absolute Value Equations and Inequalities

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Homework

• Pg 20-27, 32-35, 59, 60

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A compound statement is made up of more than one
equation or inequality. A disjunction is a
compound statement that uses the word or.
Disjunction x 3 OR x gt 2 Set builder
notation xx 3 U x gt 2
A disjunction is true if and only if at least one
of its parts is true.
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A conjunction is a compound statement that uses
the word and.
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Solving Compound Inequalities
Solve the compound inequality. Then graph the
solution set.
6y lt 24 OR y 5 3
Solve both inequalities for y.
6y lt 24
y 5 3
or
y lt 4
y 2
The solution set is all points that satisfy yy
lt 4 or y 2.
(8, 4) U 2, 8)
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Solving Compound Inequalities
Solve the compound inequality. Then graph the
solution set.
Solve both inequalities for c.
The solution set is the set of points that
satisfy both c 4 and c lt 0.
4, 0)
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Recall that the absolute value of a number x,
written x, is the distance from x to zero on
the number line. Because absolute value
represents distance without regard to direction,
the absolute value of any real number is
nonnegative.
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Absolute-value equations and inequalities can be
represented by compound statements. Consider the
equation x 3.
The solutions of x 3 are the two points that
are 3 units from zero. The solution is a
disjunction x 3 or x 3.
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The solutions of x lt 3 are the points that are
less than 3 units from zero. The solution is a
conjunction 3 lt x lt 3.
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The solutions of x gt 3 are the points that are
more than 3 units from zero. The solution is a
disjunction x lt 3 or x gt 3.
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Note The symbol can replace lt, and the rules
still apply. The symbol can replace gt, and the
rules still apply.
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Solving Absolute-Value Equations
Solve the equation.
This can be read as the distance from k to 3 is
10.
3 k 10
Rewrite the absolute value as a disjunction.
3 k 10 or 3 k 10
Add 3 to both sides of each equation.
k 13 or k 7
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Solving Absolute-Value Equations
Solve the equation.
Isolate the absolute-value expression.
Rewrite the absolute value as a disjunction.
Multiply both sides of each equation by 4.
x 16 or x 16
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Solving Absolute-Value Inequalities with
Disjunctions
Solve the inequality. Then graph the solution.
4q 2 10
Rewrite the absolute value as a disjunction.
4q 2 10 or 4q 2 10
Subtract 2 from both sides of each inequality.
4q 8 or 4q 12
Divide both sides of each inequality by 4 and
reverse the inequality symbols.
q 2 or q 3
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Solving Absolute-Value Inequalities with
Disjunctions
Solve the inequality. Then graph the solution.
0.5r 3 3
Isolate the absolute value as a disjunction.
0.5r 0
Rewrite the absolute value as a disjunction.
0.5r 0 or 0.5r 0
Divide both sides of each inequality by 0.5.
r 0 or r 0
The solution is all real numbers, R.
(8, 8)
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Solving Absolute-Value Inequalities with
Conjunctions
Solve the compound inequality. Then graph the
solution set.
Multiply both sides by 3.
2x 7 3
Rewrite the absolute value as a conjunction.
2x 7 3 and 2x 7 3
Subtract 7 from both sides of each inequality.
2x 4 and 2x 10
Divide both sides of each inequality by 2.
x 2 and x 5
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Solving Absolute-Value Inequalities with
Conjunctions
Solve the compound inequality. Then graph the
solution set.
Multiply both sides by 2, and reverse the
inequality symbol.
p 2 6
Rewrite the absolute value as a conjunction.
p 2 6 and p 2 6
Add 2 to both sides of each inequality.
p 4 and p 8
Because no real number satisfies both p 4
andp 8, there is no solution. The solution set
is ø.