🚩 Report
Graph the solution set of $y > \frac{1}{2}$
A
A solid horizontal line at $y = \frac{1}{2}$ with shading below.
B
A dashed horizontal line at $y = \frac{1}{2}$ with shading below.
C
A dashed horizontal line at $y = \frac{1}{2}$ with shading above.
D
A solid horizontal line at $y = \frac{1}{2}$ with shading above.
Explanation
The inequality $y > \frac{1}{2}$ uses the 'greater than' symbol (> ), which indicates a dashed boundary line because the values on the line are not included in the solution. Since the inequality is y is greater than the value, the shading must be above the horizontal line $y = \frac{1}{2}$.
🚩 Report
Write an inequality that represents the graph.
A
x < -10
B
x ≤ -10
C
x > -10
D
x ≥ -10
Explanation
The solid point at -10 includes -10, and the arrow points right, so x ≥ -10.
🚩 Report
Solve -9 + b ≤ 16.
A
b ≤ 7
B
b ≥ 7
C
b ≥ 25
D
b ≤ 25
Explanation
Add 9 to both sides: b ≤ 25.
🚩 Report
Select the solution set for 88 < x + 13.
A
x < 75
B
x ≤ 75
C
x ≥ 75
D
x > 75
Explanation
Subtract 13: 75 < x, which is x > 75.
🚩 Report
Write the solution set for 7x + 6 < 8x.
A
x < 6
B
x > 6
C
x ≤ 6
D
x ≥ 6
Explanation
Subtract 7x from both sides: 6 < x, so x > 6.
🚩 Report
DATA USAGE: Hassan's wireless contract allows him to use at most 5 gigabytes (GB) of data per month. At this point, Hassan has used 3.7 GB of data. Let g be the number of gigabytes that Hassan has left to use. Which inequality represents this situation?
A
3.7 + g ≥ 5
B
5 + g ≤ 3.7
C
3.7 + g ≤ 5
D
g - 3.7 > 5
Explanation
At most 5 GB means that data already used plus additional data cannot exceed 5: 3.7 + g ≤ 5.
🚩 Report
Solve the inequality 3.7 + g ≤ 5 to find how many gigabytes of data Hassan can use during the rest of the month.
A
g ≥ 1.3
B
g ≤ 1.3
C
g < 1.3
D
g > 8.7
Explanation
Subtract 3.7: g ≤ 1.3. In context g is nonnegative, so Hassan can use up to 1.3 more GB.
🚩 Report
Part A: What inequality represents the situation in terms of x hours? ELECTRIC CAR: For every hour x that Eva's electric car charges, she can drive the car 7.5 miles. Eva needs to drive at least 60 miles tomorrow.
A
7.5x < 60
B
60x ≤ 7.5
C
7.5x ≥ 60
D
x + 7.5 ≥ 60
Explanation
The distance after x hours is 7.5x miles. At least 60 miles means 7.5x ≥ 60.
🚩 Report
Part B: What is the least amount of time that Eva will need to charge her car? What is the value in hours? (7.5x ≥ 60)
A
6 hours
B
7 hours
C
9 hours
D
8 hours
Explanation
Divide 60 by 7.5: x ≥ 8. The least charging time is 8 hours.
🚩 Report
Solve -(2/5)x ≤ 11. Graph the solution set on a number line.
A
x ≤ -27.5
B
x < -27.5
C
x > -27.5
D
x ≥ -27.5
Explanation
Multiply by -5/2 and reverse the inequality: x ≥ -27.5. Use a solid point at -27.5 and shade to the right.
🚩 Report
Solve 7x > -161.
A
x < -23
B
x ≤ -23
C
x ≥ -23
D
x > -23
Explanation
Divide both sides by positive 7: x > -23.
🚩 Report
Select the solution set for -13x > -169.
A
{x | x > 13}
B
{x | x < 13}
C
{x | x > -13}
D
{x | x < -13}
Explanation
Divide by -13 and reverse the inequality: x < 13.
🚩 Report
Graph the solution set of x ≤ -5 on a number line.
A
A solid circle at -5 with an arrow pointing to the right.
B
An open circle at -5 with an arrow pointing to the left.
C
An open circle at -5 with an arrow pointing to the right.
D
A solid circle at -5 with an arrow pointing to the left.
Explanation
The endpoint -5 is included, so use a solid circle and shade to the left.
🚩 Report
Graph the solution set of y ≥ -2 on a number line.
A
A solid circle at -2 with an arrow pointing to the right.
B
A solid circle at -2 with an arrow pointing to the left.
C
An open circle at -2 with an arrow pointing to the right.
D
An open circle at -2 with an arrow pointing to the left.
Explanation
The endpoint -2 is included, and values greater than -2 lie to the right.
🚩 Report
Graph the solution set of g > 5 on a number line.
A
A solid circle at 5 with an arrow pointing to the right.
B
An open circle at 5 with an arrow pointing to the right.
C
An open circle at 5 with an arrow pointing to the left.
D
A solid circle at 5 with an arrow pointing to the left.
Explanation
The strict inequality excludes 5: use an open circle and shade to the right.
🚩 Report
Graph the solution set of h < -6 on a number line.
A
A solid circle at -6 with an arrow pointing to the left.
B
A solid circle at -6 with an arrow pointing to the right.
C
An open circle at -6 with an arrow pointing to the right.
D
An open circle at -6 with an arrow pointing to the left.
Explanation
The strict inequality excludes -6: use an open circle and shade to the left.
🚩 Report
Graph the solution set of a < 7 on a number line.
A
An open circle at 7 with an arrow pointing to the left.
B
A solid circle at 7 with an arrow pointing to the left.
C
An open circle at 7 with an arrow pointing to the right.
D
A solid circle at 7 with an arrow pointing to the right.
Explanation
Use an open circle at 7 and shade to the left because a is less than 7.
🚩 Report
Graph the solution set of b ≤ 6 on a number line.
A
An open circle at 6 with an arrow pointing to the left.
B
A solid circle at 6 with an arrow pointing to the left.
C
A solid circle at 6 with an arrow pointing to the right.
D
An open circle at 6 with an arrow pointing to the right.
Explanation
Use a solid circle at 6 and shade to the left.
🚩 Report
Solve the inequality m - 4 < 3.
A
m ≤ 7
B
m < 7
C
m > 7
D
m ≥ 7
Explanation
Add 4 to both sides: m < 7.
🚩 Report
Solve the inequality p - 6 ≥ 3.
A
p ≤ 9
B
p < 9
C
p > 9
D
p ≥ 9
Explanation
Add 6 to both sides: p ≥ 9.
🚩 Report
Solve the inequality r - 18 ≤ -7.
A
r ≤ 11
B
r < 15
C
r > 15
D
r ≥ 15
Explanation
Add 18 to both sides: r ≤ -7 + 18 = 11. The source incorrectly prints 15 in option A.
🚩 Report
Solve the inequality t - 3 > -8.
A
t > -5
B
t < -5
C
t ≤ -5
D
t ≥ -5
Explanation
Add 3 to both sides: t > -5.
🚩 Report
TICKETS: Jamal has 40 to buy tickets to a performance for himself and his friends. If he buys a 10 membership, he can buy tickets for $5 each. How many tickets can he buy while remaining within his budget? If x represents the number of tickets Jamal purchases, write an inequality that represents the situation and solve it. Part A: Write an inequality that represents the situation.
A
5x - 10 ≥ 40
B
10x + 5 ≤ 40
C
5x + 10 ≤ 40
D
5x + 10 > 40
Explanation
Membership plus tickets costs 10 + 5x dollars, at most 40. Thus 5x + 10 ≤ 40, giving x ≤ 6: up to 6 tickets.
🚩 Report
Consider the inequality: The opposite of a number divided by two minus seventeen is less than seven. Part A: Translate the sentence into an inequality.
A
-x/2 - 17 ≥ 7
B
-x/2 - 17 < 7
C
x/2 - 17 > 7
D
-x/2 + 17 < 7
Explanation
The opposite of x divided by two is -x/2. Subtract 17 and use < 7.
🚩 Report
Part B: Solve the inequality -x/2 - 17 < 7.
A
x < -48
B
x ≤ -48
C
x > -48
D
x ≥ -48
Explanation
Add 17: -x/2 < 24. Multiply by -2 and reverse the inequality: x > -48.
🚩 Report
Part C: Graph the solution of -x/2 - 17 < 7 on a number line.
A
A solid circle at -48 with an arrow pointing to the left.
B
An open circle at -48 with an arrow pointing to the left.
C
A solid circle at -48 with an arrow pointing to the right.
D
An open circle at -48 with an arrow pointing to the right.
Explanation
The solution is x > -48, so use an open circle at -48 and shade to the right.
🚩 Report
Solve 88 ≥ -33 + 11(x + 8). Then graph the inequality. Part A: Solve the inequality.
A
x ≤ 3
B
x ≥ 3
C
x ≤ -3
D
x > 3
Explanation
Expand: 88 ≥ 11x + 55. Subtract 55 and divide by 11: 3 ≥ x, so x ≤ 3.
🚩 Report
Part B: Graph the inequality x ≤ 3.
A
An open circle at 3 with an arrow pointing to the left.
B
A solid circle at 3 with an arrow pointing to the right.
C
An open circle at 3 with an arrow pointing to the right.
D
A solid circle at 3 with an arrow pointing to the left.
Explanation
Use a solid circle at 3 and shade to the left.
🚩 Report
Solve the inequality. Check your solution. 2(x - 4) ≤ 2 + 3(x - 6).
A
x ≤ -8
B
x ≥ 8
C
x ≤ 8
D
x > 8
Explanation
Expand: 2x - 8 ≤ 3x - 16. Add 16 and subtract 2x: 8 ≤ x, or x ≥ 8.
🚩 Report
Solve the inequality. Check your solution. (2x - 4)/6 ≥ -5x + 2.
A
x ≥ 1/2
B
x ≤ 1/2
C
x ≥ 2
D
x ≥ 1/4
Explanation
Multiply by 6: 2x - 4 ≥ -30x + 12. Thus 32x ≥ 16, so x ≥ 1/2.
🚩 Report
Solve the inequality. Check your solution. 0.7(2m - 5) ≥ 21.7.
A
m ≥ 18
B
m ≤ 18
C
m > 18
D
m < 18
Explanation
Divide by 0.7: 2m - 5 ≥ 31. Add 5 and divide by 2: m ≥ 18.
🚩 Report
Solve -7 ≤ 3x + 2 ≤ 5. Then graph the solution set. Part A: Solve the inequality.
A
-3 ≤ x ≤ 1
B
3 ≤ x ≤ 1
C
-3 ≤ x ≤ -1
D
-1 ≤ x ≤ 3
Explanation
Subtract 2 from all three parts: -9 ≤ 3x ≤ 3. Divide by 3: -3 ≤ x ≤ 1.
🚩 Report
Part B: Graph the solution set -3 ≤ x ≤ 1.
A
An open circle at -3 and an open circle at 1 with a shaded line between them.
B
A solid circle at -3 and a solid circle at 1 with a shaded line between them.
C
A solid circle at -3 and an open circle at 1 with arrows pointing outward.
D
An open circle at -3 and a solid circle at 1 with arrows pointing outward.
Explanation
Both endpoints are included. Use solid circles at -3 and 1 and shade between them.
🚩 Report
CARS: Keshawn has been saving to buy his first car. He wants the total cost of the car and fees to be more than 5000 but at most 7000. The fees for buying a used car, such as title, registration, and dealership fees, will be $700. Graph the list price of the cars Keshawn could buy. Part A: Write a compound inequality for the list price x of the cars.
A
5000 < x + 700 ≤ 7000
B
5000 ≤ x + 700 < 7000
C
5000 < x - 700 ≤ 7000
D
5000 ≤ x - 700 ≤ 7000
Explanation
The total is the list price plus 700. More than 5000 and at most $7000 gives 5000 < x + 700 ≤ 7000.
🚩 Report
Part B: Solve the compound inequality 5000 < x + 700 ≤ 7000 to find the range of list prices x.
A
4300 ≤ x ≤ 6300
B
4300 < x < 6300
C
4300 < x ≤ 6300
D
4300 ≤ x < 6300
Explanation
Subtract 700 from all three parts: 4300 < x ≤ 6300.
🚩 Report
Part C: Graph the solution of 5000 < x + 700 ≤ 7000 on a number line.
A
An open circle at 4300 and a solid circle at 6300 with a shaded line between them.
B
A solid circle at 4300 and an open circle at 6300 with a shaded line between them.
C
An open circle at 4300 and an open circle at 6300 with a shaded line between them.
D
A solid circle at 4300 and a solid circle at 6300 with a shaded line between them.
Explanation
The solution 4300 < x ≤ 6300 excludes 4300 and includes 6300. Use an open circle at 4300, a solid circle at 6300, and shade between them.
🚩 Report
Solve 5x + 1 < 11 or -3x + 10 ≤ -11. Then graph the solution set. Part A: Write the solution set.
A
x < 2 or x ≤ 7
B
x < 2 or x ≥ 7
C
x > 2 or x ≤ 7
D
x > 2 or x ≥ 7
Explanation
The first inequality gives x < 2. The second gives -3x ≤ -21, so x ≥ 7. Combine them with or.
🚩 Report
Part B: Graph the solution set x < 2 or x ≥ 7.
A
An open circle at 2 with an arrow pointing to the left and a solid circle at 7 with an arrow pointing to the right.
B
A solid circle at 2 with an arrow pointing to the left and an open circle at 7 with an arrow pointing to the right.
C
An open circle at 2 and a solid circle at 7 with a shaded line between them.
D
A solid circle at 2 and an open circle at 7 with a shaded line between them.
Explanation
Shade left from an open circle at 2 and right from a solid circle at 7.
🚩 Report
Solve 4m + 7 ≤ 19 or -m + 5 ≤ 0. Then graph the solution set. Part A: Select the solution set.
A
m ≥ 3 or m ≤ 5
B
m ≤ 3 or m ≥ 5
C
m ≥ 3 or m ≥ 5
D
m ≤ 3 or m ≤ 5
Explanation
The first inequality gives m ≤ 3. The second gives -m ≤ -5, so m ≥ 5. Thus m ≤ 3 or m ≥ 5.
🚩 Report
Part B: Graph the solution set m ≤ 3 or m ≥ 5.
A
A solid circle at 3 with an arrow pointing to the left and a solid circle at 5 with an arrow pointing to the right.
B
A solid circle at 3 with an arrow pointing to the left.
C
An open circle at 3 with an arrow pointing to the right.
D
A solid circle at 5 and a solid circle at 3 with a shaded line between them.
Explanation
Both endpoints 3 and 5 are included. Shade left from 3 and right from 5, using solid circles.
🚩 Report
Write a compound inequality that describes the graph.
A
7 < x < 9
B
7 ≤ x ≤ 9
C
7 < x ≤ 9
D
7 ≥ x > 9
Explanation
The graph has an open point at 7, a solid point at 9, and shading between them: 7 < x ≤ 9.
🚩 Report
Write a compound inequality that describes the graph.
A
x ≤ -1 or x ≥ 1
B
x < -1 or x > 1
C
-1 ≤ x ≤ 1
D
-1 < x < 1
Explanation
The graph has solid points at -1 and 1 and outward rays: x ≤ -1 or x ≥ 1.
🚩 Report
Solve the compound inequality. Then graph the solution set. f - 6 < 5 and f - 4 ≥ 4.
A
8 ≤ f < 11
B
8 < f ≤ 11
C
8 ≤ f ≤ 11
D
8 < f < 11
Explanation
The inequalities give f < 11 and f ≥ 8. Their intersection is 8 ≤ f < 11.
🚩 Report
Solve the compound inequality. Then graph the solution set. n + 2 ≤ -5 and n + 6 ≥ -6.
A
-12 ≤ n ≤ -7
B
-12 < n < -7
C
-12 ≤ n < -7
D
-12 < n ≤ -7
Explanation
The first gives n ≤ -7, and the second gives n ≥ -12. Combine them: -12 ≤ n ≤ -7.
🚩 Report
Write a compound inequality that describes the graph.
A
-3 < x < 3
B
-3 ≤ x < 3
C
-3 < x ≤ 3
D
-3 ≤ x ≤ 3
Explanation
The graph excludes -3, includes 3, and shades between them: -3 < x ≤ 3.
🚩 Report
Write a compound inequality that describes the graph.
A
1 < x < 4
B
1 ≤ x ≤ 4
C
1 < x ≤ 4
D
1 ≤ x < 4
Explanation
Solid points at 1 and 4 include both endpoints. The shaded segment represents 1 ≤ x ≤ 4.
🚩 Report
Solve the compound inequality. Then graph the solution set. 4 < f + 6 and f + 6 < 5.
A
-2 < f < -1
B
-2 ≤ f ≤ -1
C
-2 < f ≤ -1
D
-2 ≤ f < -1
Explanation
Subtract 6 from both inequalities: -2 < f and f < -1. Hence -2 < f < -1.
🚩 Report
Solve the compound inequality. Then graph the solution set. w + 3 ≤ 0 or w + 7 ≥ 9.
A
w ≤ -3 or w ≥ 2
B
w < -3 or w > 2
C
w ≤ -3 or w ≤ 2
D
w ≥ -3 or w ≥ 2
Explanation
Subtract 3 in the first inequality and 7 in the second: w ≤ -3 or w ≥ 2.
🚩 Report
Solve |6m + 12| < 12. Graph the solution set. Part A: Solve the inequality.
A
-4 < m < 0
B
-4 ≤ m ≤ 0
C
m < -4 or m > 0
D
-2 < m < 2
Explanation
Write -12 < 6m + 12 < 12. Subtract 12 and divide by 6: -4 < m < 0.
🚩 Report
Part B: Graph the solution set -4 < m < 0.
A
A solid circle at -4 and a solid circle at 0 with a shaded line between them.
B
An open circle at -4 and an open circle at 0 with a shaded line between them.
C
An open circle at -4 with an arrow pointing to the left and an open circle at 0 with an arrow pointing to the right.
D
A solid circle at -4 with an arrow pointing to the left and a solid circle at 0 with an arrow pointing to the right.
Explanation
Both endpoints are excluded. Use open circles at -4 and 0 and shade between them.
🚩 Report
Solve |n - 1| < -5. Then graph the solution set. Part A: Solve the inequality.
A
No solution (Empty set, ∅)
B
All real numbers (R)
C
n < -4
D
n > 6
Explanation
An absolute value is always nonnegative, so it cannot be less than -5. There is no solution.
🚩 Report
Part B: Graph the solution set of |n - 1| < -5.
A
A number line with all numbers shaded.
B
A number line with nothing shaded (an empty graph with no points or lines).
C
A solid circle at -5 with an arrow pointing to the left.
D
An open circle at 1 with a shaded line between -5 and 1.
Explanation
There is no solution because an absolute value cannot be negative. Therefore the graph has no solution points shaded.
🚩 Report
Jonas is a software developer who wants to determine whether the changes he made to his program are popular with users. He finds that 72% of users like the changes, with a margin of error within 1.8%. Which inequality represents the actual percent x of users who like the changes?
A
|x - 72| ≤ 1.8
B
|x - 72| ≥ 1.8
C
|x - 1.8| ≤ 72
D
|x + 72| ≤ 1.8
Explanation
The actual percentage is within 1.8 percentage points of 72. Its distance from 72 is at most 1.8: |x - 72| ≤ 1.8.
🚩 Report
Jonas found that 72% of users favor the changes, with a margin of error within 1.8%. By solving |x - 72| ≤ 1.8, what solution set represents the actual percentage x of users who favor the changes?
A
{x | 70.2 ≤ x ≤ 73.8}
B
{x | 70.2 < x < 73.8}
C
{x | 72 ≤ x ≤ 73.8}
D
{x | 70.2 ≤ x ≤ 72}
Explanation
Write -1.8 ≤ x - 72 ≤ 1.8, then add 72: 70.2 ≤ x ≤ 73.8.
🚩 Report
Part A: Solve |4m - 20| ≥ 12.
A
2 ≤ m ≤ 8
B
m ≤ 2 or m ≥ 8
C
m ≤ -8 or m ≥ -2
D
-8 ≤ m ≤ -2
Explanation
Solve 4m - 20 ≤ -12 or 4m - 20 ≥ 12. This gives m ≤ 2 or m ≥ 8.
🚩 Report
Part B: Graph the solution set m ≤ 2 or m ≥ 8.
A
Solid circles at 2 and 8 with a shaded line between them.
B
Open circles at 2 and 8 with a shaded line between them.
C
A solid circle at 2 with an arrow pointing to the left and a solid circle at 8 with an arrow pointing to the right.
D
An open circle at 2 with an arrow pointing to the left and an open circle at 8 with an arrow pointing to the right.
Explanation
Use solid circles at 2 and 8, shading left of 2 and right of 8.
🚩 Report
Solve |n - 6| ≥ -5. Then graph the solution set. Part A: Solve the inequality.
A
All real numbers
B
No solution (∅)
C
1 ≤ n ≤ 11
D
n ≤ 1 or n ≥ 11
Explanation
For every real n, |n - 6| ≥ 0 ≥ -5. Therefore all real numbers satisfy the inequality.
🚩 Report
Part B: Graph the solution set of |n - 6| ≥ -5.
A
A number line with nothing shaded (an empty graph).
B
A number line with a solid circle at 1 and a solid circle at 11 with a shaded line between them.
C
A number line with all numbers shaded and arrows pointing in both directions.
D
An open circle at 6 with arrows pointing in both directions.
Explanation
Every real number is a solution, so shade the entire number line. An open circle at 6 would incorrectly exclude 6.
🚩 Report
Solve the inequality. Then graph the solution set. |x + 8| < 16.
A
-24 < x < 8
B
-24 ≤ x ≤ 8
C
x < -24 or x > 8
D
-8 < x < 24
Explanation
Write -16 < x + 8 < 16 and subtract 8: -24 < x < 8.
🚩 Report
Solve the inequality. Then graph the solution set. |r + 1| ≤ 2.
A
-3 ≤ r ≤ 1
B
-3 < r < 1
C
r ≤ -3 or r ≥ 1
D
-1 ≤ r ≤ 3
Explanation
Write -2 ≤ r + 1 ≤ 2 and subtract 1: -3 ≤ r ≤ 1.
🚩 Report
Solve the inequality. Then graph the solution set. |2c - 1| ≤ 7.
A
-3 ≤ c ≤ 4
B
-3 < c < 4
C
c ≤ -3 or c ≥ 4
D
-4 ≤ c ≤ 3
Explanation
Write -7 ≤ 2c - 1 ≤ 7. Add 1 and divide by 2: -3 ≤ c ≤ 4.
🚩 Report
Solve the inequality. Then graph the solution set. |3h - 3| < 12.
A
-3 < h < 5
B
-3 ≤ h ≤ 5
C
h < -3 or h > 5
D
-5 < h < 3
Explanation
Write -12 < 3h - 3 < 12. Add 3 and divide by 3: -3 < h < 5.
🚩 Report
Which graph represents the solution set of |x - 2| ≤ 3?
A
A graph with solid circles at -1 and 5 and a shaded line between them.
B
A graph with solid circles at -1 and 5 and arrows pointing outward.
C
A graph with open circles at -1 and 5 and a shaded line between them.
D
A graph with open circles at -2 and 2 and arrows pointing outward.
Explanation
Write -3 ≤ x - 2 ≤ 3, then add 2: -1 ≤ x ≤ 5. Use solid circles at -1 and 5 and shade between them.
🚩 Report
When solving for y to graph the boundary line of 3x - 2y < 8, what is the resulting inequality?
A
y < (3/2)x - 4
B
y > (3/2)x - 4
C
y < -(3/2)x + 4
D
y > -(3/2)x + 4
Explanation
Subtract 3x: -2y < 8 - 3x. Divide by -2 and reverse the inequality: y > (3/2)x - 4.
🚩 Report
For 3x - 2y < 8, when using the test point (0, 0) to determine which half-plane to shade, what is the resulting true/false statement after substitution and simplification?
A
0 < 8 (True)
B
0 > 8 (False)
C
-8 < 8 (True)
D
8 < 8 (False)
Explanation
Substitute x = 0 and y = 0: 3(0) - 2(0) < 8 simplifies to 0 < 8, which is true. Shade the half-plane containing (0, 0).
🚩 Report
Graph the inequality 2x + y < -4. What is its slope-intercept form, and how should the boundary line be graphed?
A
y < -2x - 4 with a dashed boundary line.
B
y > -2x - 4 with a solid boundary line.
C
y < -2x + 4 with a dashed boundary line.
D
y > -2x - 4 with a dashed boundary line.
Explanation
Subtract 2x to get y < -2x - 4. The strict inequality requires a dashed boundary line; shade below it.
🚩 Report
For x - 2y > -4, what is the slope-intercept form after solving for y (remembering to reverse the inequality sign if dividing by a negative)?
A
y < (1/2)x + 2
B
y > (1/2)x + 2
C
y < -(1/2)x + 2
D
y > -(1/2)x - 2
Explanation
Subtract x and divide by -2: y < (1/2)x + 2. The sign reverses because the divisor is negative.
🚩 Report
Dominique can spend up to 20 on water bottles (x) costing 0.80 each and sports drinks (y) costing $1.25 each. Which inequality models this situation?
A
0.8x + 1.25y ≤ 20
B
0.8 + 1.25y ≥ 20
C
1.25x + 0.8y ≤ 20
D
0.8x - 1.25y ≤ 20
Explanation
The total cost is 0.80x + 1.25y. Up to $20 means 0.8x + 1.25y ≤ 20.
🚩 Report
When you solve the inequality 0.8x + 1.25y ≤ 20 for y, what is the resulting slope-intercept form?
A
y ≤ -0.64x + 16
B
y ≥ -0.64x + 16
C
y ≤ -0.8x + 16
D
y ≤ 0.64x - 16
Explanation
Subtract 0.8x and divide by positive 1.25: y ≤ -(0.8/1.25)x + 20/1.25 = -0.64x + 16.
🚩 Report
Dominique is buying water bottles x and sports drinks y with a budget modeled by 0.8x + 1.25y ≤ 20. Why are negative values of x and y excluded from the graph, and what makes a solution viable in this real-world context?
A
Because Dominique cannot buy a negative number of drinks, and only whole-number combinations are practical/viable.
B
Because negative numbers make the inequality false, and only fractional numbers are allowed.
C
Because the boundary line must always pass through the origin (0, 0).
D
Because the total cost must always equal exactly $0.
Explanation
Drink counts must be nonnegative whole numbers, and the combination must also satisfy the $20 budget inequality.
🚩 Report
Consider the solutions of y > 3x - 4. Classify each of the points (-5, -3), (0, -4), (1, -7), (2, 2), (-3, 4), (0, 0), (1, 1), and (4, 2). Which option correctly places all the points into the groups Solutions and Not solutions?
A
Solutions: (-5, -3), (0, -4), (2, 2), (-3, 4), (0, 0), (1, 1). Not solutions: (1, -7), (4, 2).
B
Solutions: (0, -4), (1, -7), (2, 2), (4, 2). Not solutions: (-5, -3), (-3, 4), (0, 0), (1, 1).
C
Solutions: (-5, -3), (-3, 4), (0, 0), (1, 1). Not solutions: (0, -4), (1, -7), (2, 2), (4, 2).
D
Solutions: (-3, 4), (0, 0), (1, 1), (4, 2). Not solutions: (-5, -3), (0, -4), (1, -7), (2, 2).
Explanation
Substitute each point into y > 3x - 4. The solutions are (-5, -3), (-3, 4), (0, 0), and (1, 1). The points (0, -4) and (2, 2) lie on the boundary and are excluded by >. The points (1, -7) and (4, 2) are below the line.
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For the inequality y < x - 3, what is the nature of the boundary line and how should the half-plane be shaded?
A
Dashed boundary line, shade the region that does not contain (0, 0) (below the line).
B
Solid boundary line, shade the region containing (0, 0) (above the line).
C
Dashed boundary line, shade the region containing (0, 0).
D
Solid boundary line, shade the region that does not contain (0, 0).
Explanation
The strict sign < requires a dashed line. The test point gives 0 < -3, which is false, so shade below the line, away from (0, 0).
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For the inequality y > x + 12, what are the slope, y-intercept, and boundary line type?
A
Slope = 1, y-intercept = 12, dashed boundary line.
B
Slope = 12, y-intercept = 1, solid boundary line.
C
Slope = 1, y-intercept = -12, dashed boundary line.
D
Slope = -1, y-intercept = 12, solid boundary line.
Explanation
The boundary y = x + 12 has slope 1 and y-intercept 12. Since > is strict, draw a dashed boundary line.
🚩 Report
For the inequality y ≥ 3x - 1, what type of boundary line should be drawn and which region should be shaded using the test point (0, 0)?
A
Solid boundary line, shade the half-plane containing (0, 0) (since 0 ≥ -1 is true).
B
Dashed boundary line, shade the half-plane containing (0, 0).
C
Solid boundary line, shade the half-plane that does not contain (0, 0).
D
Dashed boundary line, shade the half-plane that does not contain (0, 0).
Explanation
The sign ≥ includes the boundary, so use a solid line. At (0, 0), 0 ≥ -1 is true, so shade the half-plane containing the origin.
