Solving
SolvingAbsolute-Value
and Graphing Absolute-Value
2-8
2-8,9 Equations and Inequalities
Equations and Inequalities
Lesson Plan
Warm Up
Objective and PA Math Standards
Vocab
Lesson Presentation
Text Questions
Worksheet
Lesson Quiz
Holt
Algebra
Holt
Algebra
2 2
Solving
Absolute-Value
Solving
and
Graphing Absolute-Value
2-8 Equations and Inequalities
Equations and Inequalities
Warm Up
Solve.
1. y + 7 < –11 y < –18
2. 5 – 2x ≤ 17 x ≥ –6
Evaluate each expression in #3-5 for f(4) and f(-3).
3. f(x) = –|x + 1| –5; –2
4. f(x) = 2|x| – 1
7; 5
5. f(x) = |x + 1| + 2
7; 4
Use interval notation to show the graphed numbers
6.
(-2, 3]
7.
(-, 1]
Holt Algebra 2
Solving
Absolute-Value
Solving
and
Graphing Absolute-Value
2-8 Equations
andand
Inequalities
Equations
Inequalities
Objectives
Solve and graph compound
inequalities.
Write, solve and graph absolute-value
equations and inequalities.
Holt Algebra 2
Solving
Absolute-Value
Solving
and
Graphing Absolute-Value
2-8 Equations and Inequalities
Equations and Inequalities
Vocabulary
compound statement
disjunction
conjunction
absolute-value
absolute value function
Holt Algebra 2
Solving
Absolute-Value
Solving
and
Graphing Absolute-Value
2-8 Equations and Inequalities
Equations and Inequalities
A compound statement is made up of more than one
equation or inequality.
A disjunction is a compound statement using
or.
Disjunction: x ≤ –3 OR x > 2
Set builder notation: {x|x ≤ –3 U x > 2}
Interval Notation: (-∞, -2] U (2, ∞)
A disjunction is true if at least one of its parts is true.
Holt Algebra 2
Solving
Absolute-Value
Solving
and
Graphing Absolute-Value
2-8 Equations and Inequalities
Equations and Inequalities
A conjunction is a compound statement using
U
Conjunction: x ≥ –3 AND x < 2
Set builder notation: {x|x ≥ –3
Interval Notation: [-3, 2)
and.
x < 2}
A conjunction is true only if all of its parts are
true. Conjunctions can be written as a single
statement as shown.
x ≥ –3 and x< 2
–3 ≤ x < 2
Holt Algebra 2
Solving
Absolute-Value
Solving
and
Graphing Absolute-Value
2-8 Equations and Inequalities
Equations and Inequalities
Reading Math
Dis- means “apart.” Disjunctions have two pieces.
Con- means “together” Conjunctions have one piece.
Holt Algebra 2
Solving
Absolute-Value
Solving
and
Graphing Absolute-Value
2-8 Equations and Inequalities
Equations and Inequalities
Absolute value of a number x, written |x|, is the
distance from x to zero on the number line. Because it
represents distance without regard to direction, the
absolute value of any real number is always positive.
Holt Algebra 2
Solving
Absolute-Value
Solving
and
Graphing Absolute-Value
2-8 Equations and Inequalities
Equations and Inequalities
An absolute-value function is composed of two
linear pieces, one with a negative slope and one
with a positive slope.
The graph of the absolute-value “parent” function
(shown above) has a V shape with its
vertex at (0, 0) and slopes of -1 and 1.
Holt Algebra 2
Solving
Absolute-Value
Solving
and
Graphing Absolute-Value
2-8 Equations and Inequalities
Equations and Inequalities
Example 1A: Solving Compound Inequalities
Solve and graph the compound inequality.
6y < –24 OR y +5 ≥ 3
/6
Inequality Notation
Set Builder Notation
Interval Notation
/6
-5
y < –4
y ≥ –2
or
{y|y < –4 or y ≥ –2}
(–∞,-4) U [-2, ∞)
–6 –5 –4 –3 –2 –1
Holt Algebra 2
-5
0
1
2
3
Solving
Absolute-Value
Solving
and
Graphing Absolute-Value
2-8 Equations and Inequalities
Equations and Inequalities
Example 1C: Solving Compound Inequalities
Solve and graph the compound inequality.
x – 5 < –2 OR –2x ≤ –10
+5 +5
/–2
/-2
Inequality Notation
x<3
or
x≥5
Set Builder Notation {x|x < 3 or x ≥ 5}.
Interval Notation
–3 –2 –1
Holt Algebra 2
(–∞, 3) U [5, ∞)
0
1
2
3
4
5
6
Solving
Absolute-Value
Solving
and
Graphing Absolute-Value
2-8 Equations and Inequalities
Equations and Inequalities
Check It Out! Example 1b
Solve and graph the compound inequality.
2x ≥ –6 AND –x > –4
x ≥ –3
x<4
{x|x ≥ –3
x < 4}.
U
[–3, 4)
–4 –3 –2 –1 0
Holt Algebra 2
1
2
3
4
5
Solving
Absolute-Value
Solving
and
Graphing Absolute-Value
2-8 Equations and Inequalities
Equations and Inequalities
Check It Out! Example 1c
Solve and graph the compound inequality.
x – 5 < 12 OR 6x ≤ 12
x < 17
x≤2
Note: Every point that satisfies x < 17 also satisfies x < 2
{x|x < 17}.
(-∞, 17)
2
Holt Algebra 2
4
6
8
10 12 14 16 18 20
Solving
Absolute-Value
Solving
and
Graphing Absolute-Value
2-8 Equations and Inequalities
Equations and Inequalities
Example 1A Continued
The graph of g(x) = |x|– 5 is the graph of f(x) =
|x| after a vertical shift of 5 units down. The
vertex of g(x) is (0, –5).
f(x)
g(x)
Holt Algebra 2
Solving
Absolute-Value
Solving
and
Graphing Absolute-Value
2-8 Equations and Inequalities
Equations and Inequalities
Example 1B Continued
The graph of g(x) = |x + 1| is the graph
of f(x) = |x| after a horizontal shift of 1
unit left. The vertex of g(x) is (–1, 0).
f(x)
g(x)
Holt Algebra 2
Solving
Absolute-Value
Solving
and
Graphing Absolute-Value
2-8 Equations and Inequalities
Equations and Inequalities
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.
Holt Algebra 2
Solving
Absolute-Value
Solving
and
Graphing Absolute-Value
2-8 Equations and Inequalities
Equations and Inequalities
The solutions of |x| < 3 are the points that are
less than 3 units from zero. The solution is a
conjunction: –3 < x < 3.
Holt Algebra 2
Solving
Absolute-Value
Solving
and
Graphing Absolute-Value
2-8 Equations and Inequalities
Equations and Inequalities
The solutions of |x| > 3 are the points that are more
than 3 units from zero. The solution is a disjunction:
x < –3 or x > 3.
Holt Algebra 2
Solving
Absolute-Value
Solving
and
Graphing Absolute-Value
2-8 Equations and Inequalities
Equations and Inequalities
Note: The symbol ≤ can replace <, and the rules
still apply. The symbol ≥ can replace >, and the
rules still apply.
Holt Algebra 2
Solving
Absolute-Value
Solving
and
Graphing Absolute-Value
2-8 Equations and Inequalities
Equations and Inequalities
Example 2A: Solving Absolute-Value Equations
Solve the equation.
|–3 + k| = 10
–3 + k = 10 or –3 + k = –10
k = 13 or k = –7
Holt Algebra 2
Solving
Absolute-Value
Solving
and
Graphing Absolute-Value
2-8 Equations and Inequalities
Equations and Inequalities
Example 2B: Solving Absolute-Value Equations
Solve the equation.
x = 16 or x = –16
Holt Algebra 2
Solving
Absolute-Value
Solving
and
Graphing Absolute-Value
2-8 Equations and Inequalities
Equations and Inequalities
Check It Out! Example 2a
Solve the equation.
|x + 9| = 13
x + 9 = 13 or x + 9 = –13
x=4
Holt Algebra 2
or
x = –22
Solving
Absolute-Value
Solving
and
Graphing Absolute-Value
2-8 Equations and Inequalities
Equations and Inequalities
Check It Out! Example 2b
Solve the equation.
|6x| – 8 = 22
|6x| = 30
6x = 30 or 6x = –30
x=5
Holt Algebra 2
or
x = –5
Solving
Absolute-Value
Solving
and
Graphing Absolute-Value
2-8 Equations and Inequalities
Equations and Inequalities
You can solve absolute-value inequalities using
the same methods that are used to solve an
absolute-value equation.
Remember this:
“LESS THAN AND…
GREATER THAN OR.”
When the abs. value inequality is <, it’s an “AND”
When the abs. value inequality is >, it’s an “OR”
Holt Algebra 2
Solving
Absolute-Value
Solving
and
Graphing Absolute-Value
2-8 Equations and Inequalities
Equations and Inequalities
Example 3A: Solving Absolute-Value Inequalities with
Disjunctions
Solve the inequality. Then graph the solution.
|–4q + 2| ≥ 10
–4q + 2 ≥ 10
–4q ≥ 8
q ≤ –2
or
–4q + 2 ≤ –10
or
–4q ≤ –12
or
q≥3
{q|q ≤ –2 or q ≥ 3}
(–∞, –2] U [3, ∞)
Holt Algebra 2
Solving
Absolute-Value
Solving
and
Graphing Absolute-Value
2-8 Equations and Inequalities
Equations and Inequalities
Check It Out! Example 3a
Solve the inequality. Then graph the solution.
|4x – 8| > 12
4x – 8 > 12 or 4x – 8 < –12
4x > 20 or 4x < –4
x > 5 or
x < -1 or
x < –1
x>5
{x|x < –1 or x > 5}
(–∞, –1) U (5, ∞)
Holt Algebra 2
Solving
Absolute-Value
Solving
and
Graphing Absolute-Value
2-8 Equations and Inequalities
Equations and Inequalities
Example 4A: Solving Absolute-Value Inequalities with
Conjunctions
Solve the compound inequality. Then graph
the solution set.
|2x +7| ≤ 3
2x + 7 ≤ 3 and 2x + 7 ≥ –3
2x ≤ –4 and
x ≤ –2 and
Holt Algebra 2
2x ≥ –10
x ≥ –5
Solving
Absolute-Value
Solving
and
Graphing Absolute-Value
2-8 Equations and Inequalities
Equations and Inequalities
Example 4B: Solving Absolute-Value Inequalities with
Conjunctions
Solve the compound inequality. Then graph
the solution set.
|p – 2| ≤ –6
|p – 2| ≤ –6 and p – 2 ≥ 6
p ≤ –4 and
p≥8
Because no real number satisfies both p ≤ –4 and
p ≥ 8, there is no solution. The solution set is ø.
Holt Algebra 2
Solving
Absolute-Value
Solving
and
Graphing Absolute-Value
2-8 Equations and Inequalities
Equations and Inequalities
Check It Out! Example 4a
Solve the compound inequality. Then graph
the solution set.
|x – 5| ≤ 8
x – 5 ≤ 8 and x – 5 ≥ –8
x ≤ 13 and
Holt Algebra 2
x ≥ –3
Solving
Absolute-Value
Solving
and
Graphing Absolute-Value
2-8 Equations and Inequalities
Equations and Inequalities
Check It Out! Example 4b
Solve the compound inequality. Then graph
the solution set.
–2|x +5| > 10
|x + 5| < –5
x + 5 < –5 and x + 5 > 5
x < –10 and x > 0
Because no real number satisfies both x < –10 and
x > 0, there is no solution. The solution set is ø.
Holt Algebra 2
Solving
Absolute-Value
Solving
and
Graphing Absolute-Value
2-8 Equations and Inequalities
Equations and Inequalities
Lesson Quiz: Part I
Solve. Then graph the solution.
1. y – 4 ≤ –6 or 2y >8
–4 –3 –2 –1 0
{y|y ≤ –2 ≤ or y > 4}
1
2
3
4
5
2. –7x < 21 and x + 7 ≤ 6 {x|–3 < x ≤ –1}
–4 –3 –2
–1 0
1
2
3
4
5
Solve each equation.
3. |2v + 5| = 9
2 or –7
Holt Algebra 2
4. |5b| – 7 = 13
+4
Solving
Absolute-Value
Solving
and
Graphing Absolute-Value
2-8 Equations and Inequalities
Equations and Inequalities
Lesson Quiz: Part II
Solve. Then graph the solution.
5. |1 – 2x| > 7 {x|x < –3 or x > 4}
–4 –3 –2
–1 0
1
2
3
4
5
6. |3k| + 11 > 8 R
–4 –3 –2 –1
7. –2|u + 7| ≥ 16
Holt Algebra 2
0
ø
1
2
3
4
5