Name ———————————————————————
Date ————————————
BENCHMARK 3
(Chapters 5 and 6)
D. Graphing Inequalities
(pp. 49–51)
An inequality is a statement that compares unequal quantities. A compound
inequality is the intersection or union of two inequalities. The graph of an inequality
is the set of points that represents all solutions of the inequality. The graph of an
inequality in one variable is a line, line segment, or ray graphed on a number line. The
graph of an inequality in two variables is a half-plane and boundary line graphed on a
coordinate plane.
1. Graph an Inequality
EXAMPLE
To graph an
inequality in one
variable, use an
open circle for or and a closed
circle for b or r.
PRACTICE
The least expensive book at the City Bookstore costs $2. Graph an
inequality that describes prices of books at City Bookstore.
Solution:
Let P represent the price of a book at City Bookstore. The value of P must be greater
than or equal to 2. So, an inequality is P r 2.
0
1
2
3
4
5
6
Graph an inequality that describes the situation.
1. The shortest player on a basketball team is 74 inches tall.
2. A diver’s depth is at least 30 feet below sea level.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
BENCHMARK 3
D. Graphing Inequalities
3. A pitcher holds a maximum of 8 cups of water.
4. A computer has 162 megabytes of open storage space.
2. Write an Inequality Represented by a Graph
EXAMPLE
Write an inequality represented by the graph.
a.
b.
0
1
1
2
3
4
5
20
21
22
23
24
25
26
Solution:
a.
PRACTICE
The open circle means that 3 is NOT b.
a solution of the inequality. Because
the arrow points to the left, all
numbers less than 3 are solutions.
The graph represents the
inequality x < 3.
The closed circle means that 24 is a
solution of the inequality. Because the
arrow points to the right, all numbers
greater than 24 also are solutions.
The graph represents the
inequality x r 24.
Write an inequality represented by the graph.
5.
4 3 2 1
7.
0
1
2
3
0
1
2
3
4
5
6
7
6.
12 11 10 9 8 7 6 5
8.
6 5 4 3 2 1
0
1
Algebra 1
Benchmark 3 Chapters 5 and 6
49
Name ———————————————————————
Date ————————————
BENCHMARK 3
(Chapters 5 and 6)
3. Write and Graph Compound Inequalities
Vocabulary
EXAMPLE
Compound inequality Two inequalities joined by and or or. The graph of a compound
inequality with and contains only the points that the graphs of the separate inequalities
have in common. The graph of a compound inequality with or contains all the points on
the graphs of the separate inequalities.
Translate the verbal phrase into a compound inequality. Then graph the
inequality.
a.
All real numbers that are less than or b. All real numbers that are greater than
equal to 7 and greater than 1.
or equal to 4 or less than 24.
Solution:
a. 1 < x 7
0
PRACTICE
b. x 4 or x < 24
1
2
3
4
5
6
7
8
6 4 2
0
2
4
6
Translate the verbal phrase into a compound inequality. Then graph the
inequality.
9. All real numbers that are greater than 23 and less than 0.
10. All real numbers that are less than or equal to 2 or greater than or equal to 9.
12. All real numbers that are greater than or equal to 60 or less than 55.
4. Graph a Linear Inequality in Two Variables
Vocabulary
EXAMPLE
Linear inequality in two variables A statement that can be written in one of the
following forms: y mx 1 b; y r mx 1 b; y mx 1 b; or y b mx 1 b. The solution
of a linear inequality in two variables is the set of ordered pairs (x, y) that makes the
inequality a true statement.
Graph the inequality y 2x 1 1.
Solution:
Be sure that the
point you test
is not on the
boundary line. In
this example, you
could not use
(0, 1) because it is
on the boundary
line y 5 2x 1 1.
Step 1: Graph the boundary line, y 5 2x 1 1.
The inequality is , so use a dashed line.
Step 2: Test a point NOT on the boundary
line of the inequality. Test (0, 0) in
y 2x 1 1.
?
0 2(0) 1 1
01
Step 3: Shade the half-plane that contains (0, 0),
because (0, 0) is a solution of the inequality.
50
Algebra 1
Benchmark 3 Chapters 5 and 6
y
4
3
2
1
24 23 22 21
21
22
23
24
1 2 3 4 x
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
BENCHMARK 3
D. Graphing Inequalities
11. All real numbers that are less than 6 and greater than or equal to 21.
Name ———————————————————————
Date ————————————
BENCHMARK 3
(Chapters 5 and 6)
EXAMPLE
Graph the inequality 3x 1 2y 4.
Solution:
Step 1: Graph the equation 3x 1 2y 5 4.
The inequality is r, so use a solid line.
y
4
3
Step 2: Test (1, 0) in 3x 1 2y r 4.
2
1
3(1) 1 2(0) 4
?
3r4 24 23 22 21
21
22
23
24
Step 3: Shade the half-plane that does NOT
contain (1, 0), because (1, 0) is NOT
a solution of the inequality.
PRACTICE
1 2 3
4 x
Graph the inequality.
13. y 2x 1 3
14.
2x 2 y b 2
15. y 5x
16. 22x 1 3y b 26
17.
6x 1 4y r 28
18. x 1 y 10
Quiz
Graph an inequality that describes the situation.
2. Mandy hikes more than 7.5 miles.
Write an inequality represented by the graph.
3.
1
0
1
2
3
4
5
6
4.
6 5 4 3 2 1
0
1
Translate the verbal phrase into a compound inequality. Then graph the
inequality.
5. All real numbers that are less than 210 and greater than or equal to 213.
6. All real numbers that are greater than 6 or less than or equal to 0.
Graph the inequality.
7. y r 23x 2 4
8.
22x y
BENCHMARK 3
D. Graphing Inequalities
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
1. Jay reads a maximum of 20 pages.
9. x 2 2y r 4
Algebra 1
Benchmark 3 Chapters 5 and 6
51
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