2-8 Graphing Linear and Absolute Value Inequalities
Graph each inequality.
3.
1.
SOLUTION: The boundary of the graph is the graph of x + 4y = 2. Since the inequality symbol is ≤, the boundary line is solid. Test the point (0, 0).
SOLUTION: The boundary of the graph is the graph of y = 4. Since the inequality symbol is , the boundary line is solid. y is less than 4, so the region less than 4 is shaded.
The region that contains (0, 0) is shaded.
2.
SOLUTION: The boundary of the graph is the graph of x = –6. Since the inequality symbol is ≥, the boundary line is solid. x is greater than or equal to –6, so the region greater than –6 is shaded.
4.
SOLUTION: The boundary of the graph is the graph of 3x + y = – 8. Since the inequality symbol is >, the boundary line is dashed. Test the point (0, 0).
The region that contains (0, 0) is shaded.
3.
SOLUTION: The boundary of the graph is the graph of x + 4y = 2. Since the inequality symbol is ≤, the boundary line is solid. eSolutions Manual - Powered by Cognero Test the point (0, 0).
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2-8 Graphing Linear and Absolute Value Inequalities
4.
5. CCSS MODELING Gregg needs to buy gas and oil for his car. Gas costs $3.45 a gallon, and oil costs $2.41 a quart. He has $50 to spend.
SOLUTION: The boundary of the graph is the graph of 3x + y = – 8. Since the inequality symbol is >, the boundary line is dashed. Test the point (0, 0).
a. Write an inequality to represent the situation, where g is the number of gallons of gas he buys and q is the number of quarts of oil.
b. Graph the inequality. c. Can Gregg buy 10 gallons of gasoline and 8 quarts of oil? Explain.
SOLUTION: a. The inequality that represents the given situation is 3.45g + 2.41q ≤ 50. b. The boundary of the graph is the graph of 3.45g + 2.41q = 50. Since the inequality symbol is ≤, the boundary line is solid. Test the point (0, 0).
The region that contains (0, 0) is shaded.
The region that contains (0, 0) is shaded.
5. CCSS MODELING Gregg needs to buy gas and oil for his car. Gas costs $3.45 a gallon, and oil costs $2.41 a quart. He has $50 to spend.
a. Write an inequality to represent the situation, where g is the number of gallons of gas he buys and q is the number of quarts of oil.
b. Graph the inequality.
c. Can Gregg buy 10 gallons of gasoline and 8 quarts of oil? Explain.
c. The ordered pair (10, 8) is not in the shaded region. So, Gregg can’t buy 10 gallons of gasoline and 8 quarts of oil.
SOLUTION: a. The inequality that represents the given situation is 3.45g + 2.41q ≤ 50. b. The boundary of the graph is the graph of 3.45g + 2.41q = 50. Since the inequality symbol is ≤, the boundary line is solid. Test the point (0, 0).
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The region that contains (0, 0) is shaded.
Graph each inequality.
6.
SOLUTION: The boundary of the graph is the graph of y = |x + 3|. This is the absolute value graph translated 3 Page 2 units to the left. Since the inequality symbol is ≥, the boundary line is solid.
c. The ordered pair (10, 8) is not in the shaded region. 2-8 Graphing Linear So, Gregg can’t buyand 10 Absolute gallons of Value gasolineInequalities and 8 quarts of oil. Graph each inequality.
7.
6. SOLUTION: The boundary of the graph is the graph of y – 6 = |x|.
SOLUTION: The boundary of the graph is the graph of y = |x + 3|. This is the absolute value graph translated 3 units to the left. Since the inequality symbol is ≥, the boundary line is solid. Test the point (0, 0).
This is the absolute value graph translated 6 units up. Since the inequality symbol is <, the boundary line is dashed. Test the point (0, 0).
The region that contains (0, 0) is shaded. The region that does not contain (0, 0) is shaded.
Graph each inequality. 8.
7.
SOLUTION: The boundary of the graph is the graph of x + 2y = 6. Since the inequality symbol is >, the boundary line is dashed. Test the point (0, 0).
This is the absolute value graph translated 6 units up. Since the inequality symbol is <, the boundary line is dashed. Test the point (0, 0).
The region that does not contain (0, 0) is shaded.
SOLUTION: The boundary of the graph is the graph of y – 6 = |x|.
The region that contains (0, 0) is shaded.
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9.
2-8 Graphing Linear and Absolute Value Inequalities
Graph each inequality.
10.
8.
SOLUTION: The boundary of the graph is the graph of x + 2y = 6. Since the inequality symbol is >, the boundary line is dashed. Test the point (0, 0).
SOLUTION: The boundary of the graph is the graph of y = 4. Since the inequality symbol is ≤, the boundary line is solid. Test the point (0, 0).
The region that does not contain (0, 0) is shaded.
The region that contains (0, 0) is shaded.
9.
SOLUTION: The boundary of the graph is the graph of . Since the inequality symbol is ≥, the boundary line is solid. Test the point (0, 0).
11.
SOLUTION: The boundary of the graph is the graph of . Since the inequality symbol is >, the boundary line is dashed. Test the point (0, 0).
The region that contains (0, 0) is shaded.
The region that does not contain (0, 0) is shaded.
10. eSolutions Manual - Powered by Cognero
SOLUTION:
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2-8 Graphing Linear and Absolute Value Inequalities
11.
12.
SOLUTION: The boundary of the graph is the graph of . Since the inequality symbol is >, the boundary line is dashed. Test the point (0, 0).
SOLUTION: The boundary of the graph is the graph of . Since the inequality symbol is ≤, the boundary line is solid. Test the point (0, 0).
The region that does not contain (0, 0) is shaded.
The region that does not contain (0, 0) is shaded.
12.
SOLUTION: The boundary of the graph is the graph of . Since the inequality symbol is ≤, the boundary line is solid. Test the point (0, 0).
13.
SOLUTION: The boundary of the graph is the graph of . Since the inequality symbol is ≥, the boundary line is solid. Test the point (0, 0).
The region that does not contain (0, 0) is shaded.
The region that does not contain (0, 0) is shaded.
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of the test, will April be eligible for the college of her choice? SOLUTION: a. The boundary of the graph is the graph of . Since the inequality symbol is ≥, the boundary line is solid.
2-8 Graphing Linear and Absolute Value Inequalities
13.
Test the point (0, 0).
SOLUTION: The boundary of the graph is the graph of . Since the inequality symbol is ≥, the
boundary line is solid. Test the point (0, 0).
The region that does not contain (0, 0) is shaded.
Since the highest possible score in math portion is 1200, draw the inequality y ≤ 1200. Similarly the highest possible score in verbal portion is 1200, draw the inequality x ≤ 1200 in the same coordinate plane.
The region that does not contain (0, 0) is shaded.
b. The ordered pair (680, 910) is not in the shaded region. So, April is not eligible for the college of her choice.
14. COLLEGE April’s guidance counselor says that she needs a combined score of at least 1700 on her college entrance exams to be eligible for the college of her choice. The highest possible score is 2400— 1200 on the math portion and 1200 on the verbal portion.
a. The inequality represents this situation, where x is the verbal score and y is the math score. Graph this inequality. b. Refer to your graph. If she scores a 680 on the math portion of the test and 910 on the verbal portion of the test, will April be eligible for the college of her choice? SOLUTION: a. The boundary of the graph is the graph of . Since the inequality symbol is ≥, the boundary is solid. eSolutions Manualline - Powered by Cognero
Test the point (0, 0).
Graph each inequality.
15.
SOLUTION: The boundary of the graph is the graph of . This is a horizontal compression of the absolute value graph . Since the inequality symbol is >, the boundary line is dashed. Test the point (0, 0).
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b. The ordered pair (680, 910) is not in the shaded region. So, April isLinear not eligible the college of her choice. 2-8 Graphing and for Absolute Value Inequalities
Graph each inequality.
16.
15. SOLUTION: The boundary of the graph is the graph of . The equation can be written
SOLUTION: The boundary of the graph is the graph of . This is a horizontal compression of the absolute value graph . Since the inequality symbol is >, the boundary line is dashed. Test the point (0, 0).
.
This is a translation of the absolute value graph right 2 units and down 4 units. Since the inequality symbol is ≤, the boundary line is solid. Test the point (0, 0).
The region that does not contain (0, 0) is shaded.
The region that does not contain (0, 0) is shaded.
17.
16.
SOLUTION:
SOLUTION: The boundary of the graph is the graph of . The equation can be written
The related equation can be written
.
This is a translation of the absolute value graph right 2 units and down 4 units. Since the inequality symbol is ≤, the boundary line is solid. eSolutions Manual - Powered by Cognero Test the point (0, 0).
.
The negative sign reflects the absolute value graph across the y-axis (which has no effect, since it is symmetric). The 2 compresses the graph horizontally. The 6 translates the graph up 6 units.
The boundary of the graph is the graph of . Since the inequality symbol is <, the boundary line is dashed. Page 7 Test the point (0, 0).
2-8 Graphing Linear and Absolute Value Inequalities
17.
18.
SOLUTION: The related equation can be written
SOLUTION: The related equation can be written
.
The negative sign reflects the absolute value graph across the y-axis (which has no effect, since it is symmetric). The 2 compresses the graph horizontally. The 6 translates the graph up 6 units.
. The
The boundary of the graph is the graph of . Since the inequality symbol is <, the boundary line is dashed. Test the point (0, 0).
stretches the absolute value graph
horizontally. The 6 translates the graph 6 units to the left, and the –8 translates the graph 8 units down. The 2 stretches the graph vertically. The negative sign reflects the absolute value graph across the y-axis (which has no effect, since it is symmetric). The 2 compresses the graph horizontally. The 6 translates the graph up 6 units. The boundary of the graph is the graph of
. Since the inequality symbol is <,
The region that contains (0, 0) is shaded. the boundary line is dashed. Test the point (0, 0).
The region that contains (0, 0) is shaded.
18.
SOLUTION: The related equation can be written .
The
stretches the absolute value graph
horizontally. The 6 translates the graph 6 units to the left, and the –8 translates the graph 8 eSolutions Manual - Powered by Cognero units down. The 2 stretches the graph vertically. The negative sign reflects the absolute value graph
19.
SOLUTION: The related equation can be written
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.
2-8 Graphing Linear and Absolute Value Inequalities
19.
20.
SOLUTION:
SOLUTION:
The related equation can be written
.
The –5 translates the graph of 5 units to the right. The 4 compresses the graph horizontally. The compresses the graph vertically. The boundary of the graph is the graph of . Since the inequality symbol is >, the boundary line is dashed.
Test the point (0, 0).
The related equation can be written . The –4 translates the graph of 4 units to the right. The 3 compresses the graph horizontally. The negative sign in front reflects the graph across the xaxis. The boundary of the graph is the graph of . Since the inequality symbol is ≤, the boundary line is solid. Test the point (0, 0).
The region that does not contain (0, 0) is shaded.
The region that contains (0, 0) is shaded.
20.
SOLUTION: The related equation can be written . The –4 translates the graph of 4 units to the right. The 3 compresses the graph horizontally. The negative sign in front reflects the graph across the xaxis. The boundary of the graph is the graph of . Since the inequality symbol is ≤, the boundary line is solid. Manual - Powered by Cognero eSolutions Test the point (0, 0).
21. SCHOOL DANCE Carlos estimates that he will need to earn at least $700 to take his girlfriend to the prom. Carlos works two jobs as shown in the table. a. Write an inequality to represent this situation. b. Graph the inequality. c. Will he make enough working 50 hours at each job?
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situation. b. Graph the inequality. 2-8 Graphing Linear and Absolute Value Inequalities c. Will he make enough working 50 hours at each job?
Yes, Carlos will earn enough money if he works 50 hours at each job.
Graph each inequality. 22.
SOLUTION: The boundary of the graph is the graph of . The –6 translates the absolute value graph units to the right. The –2 reflects the graph across the y-axis and compresses it horizontally. Since the inequality symbol is ≥, the boundary is solid. Test the point (0, 0).
SOLUTION: a. Let a be the number of hours Carlos works at Main St. Deli. Let b be the number of hours Carlos works babysitting. The inequality that represents Carlos’s earnings is . b. The boundary of the graph is the graph of . Since the inequality symbol is ≥, the boundary is solid. Test the point (0, 0).
The region that does not contain (0, 0) is shaded.
The region that does not contain (0, 0) is shaded.
s
23.
c. Substitute 50 for x and 50 for y in the inequality .
Yes, Carlos will earn enough money if he works 50 hours at each job.
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SOLUTION: The boundary of the graph is the graph of . This is the graph of the absolute value function translated right 3 units and up 4 units. Since the inequality symbol is ≤, the boundary is solid. Page 10 Test the point (0, 0).
s 2-8 Graphing Linear and Absolute Value Inequalities 23.
24.
SOLUTION: The boundary of the graph is the graph of . This is the graph of the absolute value function translated right 3 units and up 4 units. Since the inequality symbol is ≤, the boundary is solid. Test the point (0, 0).
SOLUTION: The boundary of the graph is the graph of .
This is the graph of the absolute value function translated left 4 units, up 3 units, vertically stretched, and reflected across the x-axis. Since the inequality symbol is >, the boundary is dashed. Test the point (0, 0).
The region that contains (0, 0) is shaded.
The region that contains (0, 0) is shaded.
24.
SOLUTION: The boundary of the graph is the graph of .
25.
SOLUTION: Graph the inequality |y| > |x|. Test various values of x and y, both negative and positive, to see if they make the inequality true.
This is the graph of the absolute value function translated left 4 units, up 3 units, vertically stretched, and reflected across the x-axis. Since the inequality symbol is >, the boundary is dashed. Test the point (0, 0).
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26.
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2-8 Graphing Linear and Absolute Value Inequalities
25.
SOLUTION: Graph the inequality |y| > |x|. Test various values of x and y, both negative and positive, to see if they make the inequality true.
Therefore, this equation is true for all ordered pairs of real numbers (x, y). (The graph would be shaded everywhere.)
28. CCSS MODELING Mei is making necklaces and bracelets to sell at a craft show. She has enough beads to make 50 pieces. Let x represent the number of bracelets and y represent the number of necklaces.
a. Write an inequality that shows the possible number of necklaces and bracelets Mei can make. b. Graph the inequality. c. Give three possible solutions for the number of necklaces and bracelets that can be made.
26.
SOLUTION: Graph the inequality |x − y| > 5. Here, we want all ordered pairs for which the difference between x and y is greater than 5.
SOLUTION: a. The inequality that shows the possible number of necklaces and bracelets Mei can make is . b. The boundary of the graph is the graph of . Since the inequality symbol is ≤, the boundary is solid. Test the point (0, 0).
The region that contains (0, 0) is shaded.
27.
SOLUTION: The absolute value of any expression is always 0 or positive. So, it is always greater than any negative number. Therefore, this equation is true for all ordered pairs of real numbers (x, y). (The graph would be shaded everywhere.)
28. CCSS MODELING Mei is making necklaces and bracelets to sell at a craft show. She has enough beads to make 50 pieces. Let x represent the number eSolutions Manual - Powered by Cognero of bracelets and y represent the number of necklaces.
c. Sample answers: 0 bracelets and 50 necklaces, 25 necklaces and 25 bracelets, or 30 bracelets and 20 necklaces.
29. GIFT CARDS Susan received a gift card from an electronics store for $400. She wants to spend the money on DVDs, which cost $20 each, and CDs, which cost $15 each.
a. Let d equal the number of DVDs, and let c equal Page 12 the number of CDs. Write an inequality that shows the possible combinations of DVDs and CDs that Susan can purchase.
c. Sample answers: 0 bracelets and 50 necklaces, 25 necklaces and 25 bracelets, or 30 bracelets and 20 necklaces. Linear and Absolute Value Inequalities 2-8 Graphing
c. Sample answers: 18 CDs and 5 DVDs, 12 CDs and 10 DVDs, or 6 CDs and 15 DVDs.
29. GIFT CARDS Susan received a gift card from an electronics store for $400. She wants to spend the money on DVDs, which cost $20 each, and CDs, which cost $15 each.
Graph each inequality.
30.
SOLUTION:
a. Let d equal the number of DVDs, and let c equal the number of CDs. Write an inequality that shows the possible combinations of DVDs and CDs that Susan can purchase.
Graph the inequality
.
b. Graph the inequality.
c. Give three possible solutions for the number of DVDs and CDs she can buy.
SOLUTION:
a. The inequality that shows the possible number of DVDs and CDs Susan can purchase is . b. The boundary of the graph is the graph of . Since the inequality symbol is ≤, the boundary is solid.
31.
SOLUTION: Graph the inequality
. The boundary line
is a transformation of the graph of the left.
Test the point (0, 0).
2 units to
The region that contains (0, 0) is shaded.
32.
SOLUTION: Graph the inequality
.
c. Sample answers: 18 CDs and 5 DVDs, 12 CDs and 10 DVDs, or 6 CDs and 15 DVDs.
eSolutions Manual - Powered by Cognero Graph each inequality.
30.
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2-8 Graphing Linear and Absolute Value Inequalities
35. ERROR ANALYSIS Paulo and Janette are graphing Is either of them correct? Explain your reasoning.
32.
SOLUTION: Graph the inequality
.
33. OPEN ENDED Create an absolute value inequality in which none of the possible solutions fall in the second or third quadrant.
SOLUTION: Sample answer:
34. CHALLENGE Graph the following inequality.
SOLUTION: Paulo’s graph is correct. Test the point (0, 0).
SOLUTION: Draw each piece of the graph as a dashed line (since the main inequality is >). Shade the region above the piecewise-defined boundary.
The region that does not contain (0, 0) is to be shaded.
36. REASONING When will it be possible to shade two different areas when graphing a linear absolute value inequality? Explain your reasoning.
35. ERROR ANALYSIS Paulo and Janette are graphing Is either of them correct? eSolutions Manual - Powered by Cognero Explain your reasoning.
SOLUTION: Sample answer: It will be possible when we have a situation where y and x are both inside an absolute value. An example is When this happens, positive and negative values of y Page 14 will need to be considered, as well as the positive and negative values of x.
The region that does not contain (0, 0) is to be shaded. Linear and Absolute Value Inequalities 2-8 Graphing
36. REASONING When will it be possible to shade two different areas when graphing a linear absolute value inequality? Explain your reasoning.
SOLUTION: Sample answer: It will be possible when we have a situation where y and x are both inside an absolute value. An example is When this happens, positive and negative values of y will need to be considered, as well as the positive and negative values of x.
37. WRITING IN MATH Describe a situation in which there are no solutions to an absolute value inequality. Explain your reasoning.
SOLUTION: Sample answer: One possibility is when In order for there to be a solution, the absolute value of y will need to be less than 0, and, by definition of absolute value, this is impossible.
worth twice as much, then it is worth 200, so overall he has 600 possible points. In order to earn a 90%, he needs or 540 points. So, Craig needs 540 – 341 or 199 out of 200 on the last test: 99.5%.
39. Which of the following sets of numbers represents an infinite set?
A {2, 4, 6}
B {whole numbers between –50 and 50} C {integers}
D
SOLUTION: The set of integers is an infinite set. So, the correct choice is C.
40. SHORT RESPONSE Which theorem of congruence should be used to prove
38. EXTENDED RESPONSE Craig scored 85%, 96%, 79%, and 81% on his first four math tests. He hopes to score high enough on the final test to earn a 90% average. If the final test is worth twice as much as one of the other tests, determine if Craig can earn a 90% average. If so, what score does Craig need to get on the final test to accomplish this? Explain how you found your answer.
SOLUTION: Sample answer: Craig can earn a 90% average by scoring a 99.5% or higher on his last test. This was determined by finding the sum of the first 4 tests. By making each original test worth 100 points, then so far he has 341 out of 400 points. If the last test is worth twice as much, then it is worth 200, so overall he has 600 possible points. In order to earn a 90%, he needs or 540 points. So, Craig needs 540 – 341 or 199 out of 200 on the last test: 99.5%.
39. Which of the following sets of numbers represents an infinite set? eSolutions Manual - Powered by Cognero
A {2, 4, 6}
SOLUTION:
By SAS theorem of congruence we can prove
41. ACT/SAT For which function is the range
F G H J K
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, is a translation left or right.
By SAS theorem of congruence we can prove 2-8 Graphing Linear and Absolute Value Inequalities
So, the equation of the graph is
41. ACT/SAT For which function is the range
F G H J K
43.
SOLUTION: The given graph is a combination of transformations of the parent graph .
When a constant k is added to or subtracted from a parent function, the result is a translation of the graph up or down.
SOLUTION: The range of the function So, the correct choice is K.
is
When a constant h is added to or subtracted from x before evaluating a parent function, the result, , is a translation left or right. The graph was moved 5 units down and 4 units left. So, the equation of the graph is
Write an equation for each graph.
42.
SOLUTION: The given graph is a combination of transformations of the parent graph . When a constant k is added to or subtracted from a parent function, the result is a translation of the graph up or down. When a constant h is added to or subtracted from x before evaluating a parent function, the result, , is a translation left or right.
44.
SOLUTION: The given graph is a combination of transformations and reflection of the parent graph . When a parent function is multiplied by –1, the result is a reflection of the graph in the x-axis. The graph was moved 4 units up and 2 units right. So, the equation of the graph is .
So, the equation of the graph is
Graph each function.
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45.
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graph in the x-axis. The graph was moved 4 units up and 2 units right. 2-8 Graphing Linear and Absolute Value Inequalities So, the equation of the graph is .
Graph each function.
47.
45.
SOLUTION: Graph each line on its domain. Use solid dots for endpoints where there is a ≤ or ≥ symbol, and an open circle where there is a < or > symbol.
SOLUTION: Graph each line on its domain. Use solid dots for endpoints where there is a ≤ or ≥ symbol, and an open circle where there is a < or > symbol.
Write each equation in standard form. Identify A , B, and C.
48. –6y = 8x – 3
46.
SOLUTION: Rewrite the equation in the standard form
SOLUTION: Graph each line on its domain. Use solid dots for endpoints where there is a ≤ or ≥ symbol, and an open circle where there is a < or > symbol.
So, A = 8, B = 6 and C = 3.
49. 12y + x = –3y + 5x – 6
SOLUTION: Rewrite the given equation in the standard form
47.
So, A = 4, B = –15 and C = 6.
SOLUTION: Graph each line on its domain. Use solid dots for eSolutions Manual - Powered by Cognero endpoints where there is a ≤ or ≥ symbol, and an open circle where there is a < or > symbol.
50.
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So, A = 8, B = 6 and C = 3. 2-8 Graphing Linear and Absolute Value Inequalities 49. 12y + x = –3y + 5x – 6
Multiply.
SOLUTION: Rewrite the given equation in the standard form
52. (3x – 4)(2x + 1)
SOLUTION: Use the FOIL method to multiply two binomials.
So, A = 4, B = –15 and C = 6.
53. (6x + 5)(–x – 3)
50.
SOLUTION: Use the FOIL method to multiply two binomials.
SOLUTION: Rewrite the given equation in the standard form.
54. (5x + 2)(–2x + 3)
SOLUTION: Use the FOIL method to multiply two binomials.
So, A=1 B=2 C = 11.
51. TENNIS Sixteen players signed up for tennis lessons. The instructor plans to use 50 tennis balls for every player and have 200 extra. How many tennis balls are needed for the lessons? SOLUTION: Total number of tennis balls needed for the lessons
Graph each linear equation.
55. y = 2x – 8
SOLUTION: Graph a line with slope 2 and y-intercept –8.
Multiply.
52. (3x – 4)(2x + 1)
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SOLUTION: Use the FOIL method to multiply two binomials.
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2-8 Graphing Linear and Absolute Value Inequalities
Graph each linear equation.
57. 3y – 4x = 24
55. y = 2x – 8
SOLUTION: Graph a line with slope 2 and y-intercept –8.
SOLUTION: Graph the equation 3y – 4x = 24. The y-intercept is 8 and the x-intercept is –6.
56.
SOLUTION: Graph a line with slope
and y-intercept 2.
57. 3y – 4x = 24
SOLUTION: Graph the equation 3y – 4x = 24. The y-intercept is 8 and the x-intercept is –6.
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