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Graphing Inequalities in Two Variables12-6
Learn to graph inequalities on the coordinate plane.
Graphing Inequalities in Two Variables12-6
Graph each inequality.
y < x – 1
Example 1: Graphing Inequalities
First graph the boundary line y = x – 1. Since no points that are on the line are solutions of y < x – 1, make the line dashed. Then determine on which side of the line the solutions lie.
(0, 0)
y < x – 1
Test a point not on the line.
Substitute 0 for x and 0 for y.0 < 0 – 1?
0 < –1?
Graphing Inequalities in Two Variables12-6
Any point on the line y = x 1 is not a solution of y < x 1 because the inequality symbol < means only “less than” and does not include “equal to.”
Helpful Hint
Graphing Inequalities in Two Variables12-6
Example 1 Continued
(0, 0)
Since 0 < –1 is not true, (0, 0) is not a solution of y < x – 1. Shade the side of the line that does not include (0, 0).
Graphing Inequalities in Two Variables12-6
y 2x + 1
Example 2: Graphing Inequalities
First graph the boundary line y = 2x + 1. Since points that are on the line are solutions of y 2x + 1, make the line solid. Then shade the part of the coordinate plane in which the rest of the solutions of y 2x + 1 lie.
(0, 4) Choose any point not on the line.
Substitute 0 for x and 4 for y.
y ≥ 2x + 1
4 ≥ 0 + 1?
Graphing Inequalities in Two Variables12-6
Any point on the line y = 2x 1 is a solution of y ≥ 2x 1 because the inequality symbol ≥ means “greater than or equal to.”
Helpful Hint
Graphing Inequalities in Two Variables12-6
Example 2 Continued
Since 4 1 is true, (0, 4) is a solution of y 2x + 1. Shade the side of the line that includes (0, 4).
(0, 4)
Graphing Inequalities in Two Variables12-6
2y + 5x < 6
Example 3: Graphing Inequalities
First write the equation in slope-intercept form.
2y < –5x + 6
2y + 5x < 6
y < – x + 352
Then graph the line y = – x + 3. Since points that
are on the line are not solutions of y < – x + 3,
make the line dashed. Then determine on which
side of the line the solutions lie.
52 5
2
Subtract 5x from both sides.
Divide both sides by 2.
Graphing Inequalities in Two Variables12-6
Example 3 Continued
Since 0 < 3 is true, (0, 0) is a
solution of y < – x + 3.
Shade the side of the line
that includes (0, 0).
52
(0, 0) Choose any point not on the line.
y < – x + 352
0 < 0 + 3?
0 < 3?
Substitute 0 for x and 0 for y.
(0, 0)
Graphing Inequalities in Two Variables12-6
Graph each inequality.
y < x – 4
Example 4
First graph the boundary line y = x – 4. Since no points that are on the line are solutions of y < x – 4, make the line dashed. Then determine on which side of the line the solutions lie.
(0, 0)
y < x – 4
Test a point not on the line.
Substitute 0 for x and 0 for y.0 < 0 – 4?
0 < –4?
Graphing Inequalities in Two Variables12-6
Example 4 Continued
(0, 0)
Since 0 < –4 is not true, (0, 0) is not a solution of y < x – 4. Shade the side of the line that does not include (0, 0).
Graphing Inequalities in Two Variables12-6
y > 4x + 4
Example 5
First graph the boundary line y = 4x + 4. Since points that are on the line are solutions of y 4x + 4, make the line solid. Then shade the part of the coordinate plane in which the rest of the solutions of y 4x + 4 lie.
(2, 3) Choose any point not on the line.
Substitute 2 for x and 3 for y.
y ≥ 4x + 4
3 ≥ 8 + 4?
Graphing Inequalities in Two Variables12-6
Example 5 Continued
Since 3 12 is not true, (2, 3) is not a solution of y 4x + 4. Shade the side of the line that does not include (2, 3).
(2, 3)
Graphing Inequalities in Two Variables12-6
3y + 4x 9
Example 6
First write the equation in slope-intercept form.
3y –4x + 9
3y + 4x 9
y – x + 343
Subtract 4x from both sides.
Divide both sides by 3.
43Then graph the line y = – x + 3. Since points that
are on the line are solutions of y – x + 3, make
the line solid. Then determine on which side of the
line the solutions lie.
43
Graphing Inequalities in Two Variables12-6
Example 6 Continued
Since 0 3 is not true, (0, 0) is
not a solution of y – x + 3.
Shade the side of the line that
does not include (0, 0).
43
(0, 0) Choose any point not on the line.
y – x + 343
0 0 + 3?
0 3?
Substitute 0 for x and 0 for y.
(0, 0)
Graphing Inequalities in Two Variables12-6
Keith has $500 in a savings account at the beginning of the summer. He wants to have at least $200 in the account by the end of the summer. He withdraws $25 a week for spending money. How many weeks can Keith withdraw money in his account and still have at least $200 in at the end of summer?
Example 7: Real World
Graphing Inequalities in Two Variables12-6
The phrase “no more” can be translated as less than or equal to.
Helpful Hint
Graphing Inequalities in Two Variables12-6
Example 8: Real World
A taxi charges a $1.75 flat rate fee in addition to $0.65 per mile. Katie has no more than $15 to spend. How many miles can Katie travel without going over what she has to spend?
Graphing Inequalities in Two Variables12-6
Standard Lesson Quiz
Lesson Quizzes
Lesson Quiz for Student Response Systems
Graphing Inequalities in Two Variables12-6
Graph each inequality.
1. y < – x + 4
13
Lesson Quiz Part I
Graphing Inequalities in Two Variables12-6
2. 4y + 2x > 12
Lesson Quiz Part II
Graphing Inequalities in Two Variables12-6
1. Identify the graph of the given inequality. 6y + 3x > 12
A. B.
Lesson Quiz for Student Response Systems
Graphing Inequalities in Two Variables12-6
2. Tell which ordered pair is a solution of the inequality y < x + 12.
A. (–3, 5)
B. (–4, 12)
C. (–5, 8)
D. (–7, 9)
Lesson Quiz for Student Response Systems