Chapter 4 Represent And Solve Equations And Inequalities
Section 4.7: Solve Inequalities
Page 217 Exercise 1 Answer
Let n be the number of which Henry is thinking. It is less than 17. The inequality which shows this is:
n < 17.
To show all the numbers that are less than 17 on a number line, mark 17 on the number line with an open circle since it is not included, and mark the line on the left.
Result
n < 17
Page 217 Exercise 1 Answer
Henry could not be thinking of 17, since it clearly states he is thinking of a number less than 17.
Result
No.
Read And Learn More: enVisionmath 2.0 Grade 6 Volume 1 Solutions
Page 218 Exercise 1 Answer
To graph x < 8, start by drawing an open circle at 8 on the number line since 8 is not included as a solution.
7 and 4 are two of the many possible solutions of the inequality and both of these numbers lie to the left of 8 on the number line. To complete the graph, you then need to shade the solutions to the left of the open circle you drew at 8.
If the less than sign is changed to a greater than sign, then the only part of the graph that changes is the part you shade. Instead of shading to the left of 8, you would have to shade to the right of 8. The graph would still have an open circle at 8 since 8 would still not be included as a solution.
Result
Draw an open circle at 8 on the number line. Shade to the left of the open circle you drew at 8.
If the less than sign is changed to a greater than sign, then the only part of the graph that changes is the part you shade. Instead of shading to the left of 8, you would have to shade to the right of 8. The graph would still have an open circle at 8 since 8 would still not be included as a solution.
Page 219 Exercise 2 Answer
All the possible costs of the items on the children’s menu are marked x and for them the inequality x ≤ $8.50, is true. Also, they are obviously greater than zero.
To mark this on a number line, first mark the point 8.5 and than all the points to the left of it.
Result
x ≤ $8.50
Page 219 Exercise 3 Answer
First, write an inequality which represents the situation.
Let y mark how far a jumper has jumped. Since it states that to qualify for the finals one must jump at least \(20 \frac{1}{2}\) feet, all that qualify must have jumped either exacatly \(20 \frac{1}{2}\) feet or more.
y ≥ \(20 \frac{1}{2}\)
Check for which jumpers the inequality is true. Those are the one who would qualify.
Amir: \(22 \frac{1}{3} \geq 20 \frac{1}{2}\)
Jake: \(16 \nsucceq 20 \frac{1}{2}\)
Tyrell: \(18 \frac{1}{2} \nsucceq 20 \frac{1}{2}\)
Ryan: \(20 \frac{1}{2} \geq 20 \frac{1}{2}\)
Amir and Ryan would qualify for finals.
Result
Amir and Ryan would qualify.
Page 220 Exercise 1 Answer
You can represent the solutions of an inequality by making a graph on a number line. If the inequality symbol is < or >, plot an open circle on the number line and if it is ≤ or ≥, plot a closed circle on the number line. Then shade to the left if the symbol is < or ≤ and shade to the right if the symbol is > or ≥.
Page 220 Exercise 2 Answer
The inequality x > 5 means that x can be any number that is larger than 5. Since 9 is larger than 5, then 9 is a solution of x > 5.
Page 220 Exercise 3 Answer
The inequality x > 5 means that x can be any number that is larger than 5. Since 2 is not larger than 5, then 2 is not a solution of x > 5.
Page 220 Exercise 4 Answer
The inequality x > 12 means that x can be any number that is greater than 12, such as 12.1, \(15 \frac{1}{2}\), and 50. Since there are infinitely many numbers that are greater than 12, then the inequality has infinitely many solutions.
Result
infinitely many solutions
Page 220 Exercise 5 Answer
When the inequality symbol is < or >, we use an open circle and when the symbol is ≤ or ≥, we use a closed circle. We shade to the right when the symbol is > or ≥ and shade to the left when the symbol is < or ≤.
The graphs of the solutions of inequalities involving > and ≥ will then both be shaded to the right, but the > graph will have an open circle and the ≥ graph will have a closed circle.
Result
The graphs of the solutions of inequalities involving > and ≥ will both be shaded to the right, but the > graph will have an open circle and the ≥ graph will have a closed circle.
Page 220 Exercise 6 Answer
To determine which inequality symbol to choose, we need to look at what type of circle there is on the number line and what direction the graph is shaded.
The graph has an open circle at 14 so the inequality symbol will be either > or <.
The graph is shaded to the left of the open circle which means z must be smaller than 14. The inequality is then z < 14.
Result
z < 14
Page 220 Exercise 7 Answer
To determine which inequality symbol to choose, we need to look at what type of circle there is on the number line and what direction the graph is shaded.
The graph has a closed circle at 18 so the inequality symbol will be either ≥ or ≤.
The graph is shaded to the right of the closed circle which means d could be greater than 18. The inequality is then d ≥ 18.
Result
d ≥ 18
Page 220 Exercise 8 Answer
Use substitution to answer the question.
4.3 < 8
5.3 < 8
8.3 > 8
9 > 8
The solution to the inequality are w = 4.3 and w = 5.3, however w = 8.3 and w = 9 are not solutions.
Result
w = 4.3, w = 5.3
Page 220 Exercise 9 Answer
Use substitution to answer the question.
24 < 25
25 = 25
25.1 > 25
27 > 25
The solution to the given inequality are t = 25.1 and t = 27, however t = 24 and t = 25 are not solutions.
Result
t = 25.1, t = 27
Page 220 Exercise 10 Answer
Use substitution to find the answer the question.
0 ≤ 4
4 ≤ 4
5 > 4
6 > 4
The solution to the give inequality are g = 0 and g = 4, however g = 5 and g = 6 are not solutions.
Result
g = 0, g = 4
Page 220 Exercise 11 Answer
Use substitution to find the solution of the inequality.
4 < 8
5 < 8
6 < 8
7 < 8
Neither of the given values of the variable is the solution.
Result
Neither of the given values of the variable is the solution.
Page 221 Exercise 12 Answer
Number 7 beginning of the marked segment on the number line, however the point 7 itself is not marked. The marked segment is to the left of 7. Thus, the inequality that the graph represents is
y < 7.
Result
y < 7
Page 221 Exercise 13 Answer
Number 0 is the beginning of the marked segment on the number line, however the point 0 itself is not marked. The marked segment is to the right of 0. Thus, the inequality that the graph represents is
b > 0.
Result
b > 0
Page 221 Exercise 14 Answer
Number 3 is the beginning of the marked segment on the number line, however the point 3 itself is not marked, so x ≠ 3. The marked segment is to the right of 3, so all x are greater than 3. Thus, the inequality that the graph represents is
x > 3.
Result
x > 3
Page 221 Exercise 15 Answer
Number 5 is the beginning of the marked segment on the number line, and since the point 5 itself is is marked, t = 5. The marked segment is to the left of the point 5, so the all t are less than 5. Thus, the inequality that the graph represents is
t ≤ 5.
Result
t ≤ 5
Page 221 Exercise 16 Answer
To graph the inequality on a number line, first mark the point 9 with a closed circle. Since the inequality states that h are all numbers equal to or greater than 9, mark the line segment to the right of the point 9.
Result
Mark the point 9 with a closed circle and mark the line segment to the right of the point 9.
Page 221 Exercise 17 Answer
The inequality states that p are all numbers less than 3. First mark the point 3, however, since p ≠ 3, mark it only with an open circle. Mark the line to the left of the point 3.
Result
Mark the point 3 with an open circle and mark the line to the left of the point 3.
Page 221 Exercise 18 Answer
The inequality states that t are all the numbers equal to or less than 6, thus mark the point 6 with a closed circle and the line to the left of the point 6.
Result
Mark the point 6 with a closed circle and mark the line to the left of the point 6.
Page 221 Exercise 19 Answer
The inequality states that the s are all the numbers greater than 1, thus mark the point 1 with an open circle and mark the line to the right of the point 1.
Result
Mark the point 1 with an open circle and mark the line to the right of the point 1.
Page 221 Exercise 20 Answer
The solutions to the inequality are all numbers greater than 10.5, for example: 11, 321, or 10.9.
Result
11, 321, 10.9
Page 221 Exercise 21 Answer
The solutions to the equation are all the numbers less than 19. For example: 18, 1.3, and −9.
Result
18, 1.3, −9
Page 221 Exercise 22 Answer
The solutions to the inequality are all the numbers equal to or greater than 200. For example: 540, 201.34, and 200.
Result
540, 201.34, 200
Page 221 Exercise 23 Answer
The solutions to the inequality are all the numbers equal to or less than 82. For example: 82, −4, and 81.4.
Result
82, −4, 81.4
Page 221 Exercise 24 Answer
The solutions to the inequality are all the numbers equal to or greater than 12. For example: 12, 321, and 12.23.
Result
12, 321, 12.23
Page 221 Exercise 25 Answer
The solutions to the inequality are all the numbers equal to or less than 3.5. For example: 2, −3.4, and 3.5.
Result
2, −3.4, 3.5
Page 221 Exercise 26 Answer
The solutions to the inequality are all the numbers grater than 35. For example: 38, 35.4, and 342.
Result
38, 35.4, 342
Page 221 Exercise 27 Answer
The solutions to the inequality are all the numbers which are less than 2.5. For example: −2, 2, and 2.4.
Result
−2, 2, 2.4
Page 221 Exercise 28 Answer
Substitute w with 1,505 and 1,600 in the inequality and check wheather it is true.
w ≤ 1,500
1,505 ≥ 1,500
1,600 ≥ 1,500
Since neither of the inequalities are true when w is substituted, 1,505 pounds and 1,600 pounds are not allowed in a freight elevator.
Result
No.
Page 222 Exercise 29 Answer
In the inequality x > 2, the solution are all numbers greater than 2. In the inequality x < 2, the solution are all the numbers less than 2. Notice, 2 is not a solution to either of the inequalities.
Result
2 is not a solution to either of the inequalities.
Page 222 Exercise 30 Answer
Let t mark the temperature at the Death Valley.
Since the highest recorded temperature was 134° F, the inequality showing this is:
t ≤ 134°F.
The lowest recorded temperature was 15°F, thus the inequality showing this is:
t ≥ 15°F.
Result
t ≤ 134° F, t ≥ 15°F.
Page 222 Exercise 31 Answer
The solutions of the inequality x > 7 are all the numbers greater than 7, thus both 7.1 and 7.01 are solutions since both of them are greater, even though by a small difference, but still they are greater. That is, if we look at the number line they are to the right of the point 7 and are thus solutions.
Result
Both 7.1 and 7.01 are solutions of the inequality.
Page 222 Exercise 32 Answer
Since the temperature should be either exactly 65 degrees or higher the inequality describing the situation is:
t ≥ 65°,
where t represents the allowable temperature in the greenhouse.
Result
t ≥ 65°
Page 222 Exercise 33 Answer
Since the gift card’s value is enough to buy any of the apps thus it must be equal to or greater than the most expensive app.
$9.50 < $10.50 < $12.00
The most expensive app is the Remote Desktop which costs $12.00. Thus, the value of the gift card must be at least $12.00. If the value of the gift card is markedv, the inequality which represents this situation is:
v ≥ $12.00.
Result
v ≥ $12.00
Page 222 Exercise 34 Answer
Since 400 pounds is the maximum weight, w, the weight on the plane must be either equal to or less than 400 pounds.
w ≤ 400.
Result
w ≤ 400
Page 222 Exercise 35 Answer
Inequality n > 21 is true for all numbers greater than 21, that is, n = 22, 23, 24, 25, …
Result
n = 22, 23, 24, 25, …
Page 222 Exercise 36 Answer
The first two tick marks on the number line are 3.0 and 3.1 so we know the number line has an interval of 0.1. Finish labeling the number line with 3.2 through 3.9.
To graph the inequality y < 3.7, we need to start by plotting an open circle at 3.7 since the inequality symbol doesn’t have an equal sign. Since y must be a number smaller than 3.7, we then need to shade to the left of 3.7 on the number line:
Result
Finish labeling the number line with 3.2 through 3.9. Plot an open circle at 3.7 and then shade to the left.
Page 222 Exercise 37 Answer
The first two tick marks on the number line are 20 and 21 so we know the number line has an interval of 1. Finish labeling the number line with 23 through 28.
To graph the inequality x ≤ 25, we need to start by plotting a closed circle at 25 since the inequality symbol has an equal sign. Since x can be a number smaller than 25, we then need to shade to the left of 25 on the number line:
Result
Finish labeling the number line with 23 through 28. Plot a closed circle at 25 and then shade to the left.