Math: Systems of Inequalities Worksheet

Question 1

Solve the system and describe the solution set: y > 2x - 1 and y <= x + 3.

Accepted Answer

The solution set is all points that lie above the line y = 2x - 1 and on or below the line y = x + 3. These points satisfy both inequalities at the same time.

Question 2

Graph the system: x >= -2 and y < 4. Describe the region that is shaded.

Accepted Answer

The solution set is the region on or to the right of the vertical line x = -2 and below the horizontal line y = 4. The line x = -2 is solid, and the line y = 4 is dashed.

Question 3

Determine whether the point (2, 1) is a solution to the system: y < x and y >= -x + 2.

Accepted Answer

Yes, the point (2, 1) is a solution. When x = 2 and y = 1, the inequality 1 < 2 is true and the inequality 1 >= 0 is also true.

Question 4

Determine whether the point (-1, 3) is a solution to the system: y <= 2x + 5 and y > -x + 1.

Accepted Answer

Yes, the point (-1, 3) is a solution. Substituting gives 3 <= 3, which is true, and 3 > 2, which is also true.

Question 5

Write the system of inequalities represented by this description: the region above the line y = -3x + 2 and on or below the line y = x - 1.

Accepted Answer

The system is y > -3x + 2 and y <= x - 1. The first inequality is strict because the region is above the line, and the second includes the boundary because it is on or below the line.

Question 6

A school club sells notebooks for x dollars and pens for y dollars. The prices must satisfy x >= 1, y >= 0.5, and x + y <= 5. Describe what the solution set represents.

Accepted Answer

The solution set represents all possible notebook and pen prices that meet the minimum price rules and keep the total of one notebook and one pen at no more than 5 dollars. It is the region in the first quadrant on or above the minimum values and on or below the line x + y = 5.

Question 7

Solve the system: y < -2x + 6 and y > -2x + 1. Describe the region between the lines.

Accepted Answer

The solution set is all points between the parallel lines y = -2x + 6 and y = -2x + 1. Points must be below the first line and above the second line, so the shaded region is the strip between them, not including either boundary.

Question 8

Graph and describe the solution set for the system: y >= 3 and y < -x + 7.

Accepted Answer

The solution set is all points on or above the horizontal line y = 3 and below the line y = -x + 7. The overlap forms a region that includes the line y = 3 and excludes the line y = -x + 7.

Question 9

Find one point that satisfies the system y > x - 4 and y < 2x + 1. Explain why it works.

Accepted Answer

One possible point is (2, 0). It works because 0 > 2 - 4, so 0 > -2 is true, and 0 < 2(2) + 1, so 0 < 5 is also true.

Question 10

Is there a solution to the system y > x + 2 and y < x - 1. Explain your reasoning.

Accepted Answer

No, there is no solution. The line y = x + 2 is always 3 units above the line y = x - 1, so a point cannot be above the higher line and below the lower line at the same time.

Question 11

Write a system of two inequalities whose solution is the region in the first quadrant below the line y = -x + 6.

Accepted Answer

One correct system is x >= 0, y >= 0, and y <= -x + 6. These inequalities restrict the region to the first quadrant and keep all points on or below the line.

Question 12

A student must spend at least 4 hours on math, x, and science, y, combined, so x + y >= 4. The student also wants to spend no more than 6 hours total, so x + y <= 6, with x >= 0 and y >= 0. Describe the feasible region.

Accepted Answer

The feasible region is the set of all nonnegative points between the lines x + y = 4 and x + y = 6 in the first quadrant. It includes both boundary lines because the inequalities are inclusive.

Question 13

For the system y <= -x + 5 and y >= 2, find two ordered pairs that are solutions.

Accepted Answer

Two possible solutions are (1, 3) and (2, 2). For (1, 3), 3 <= 4 and 3 >= 2 are both true. For (2, 2), 2 <= 3 and 2 >= 2 are both true.

Question 14

Explain the difference between graphing y < 2x + 1 and y <= 2x + 1 in a system of inequalities.

Accepted Answer

For y < 2x + 1, the boundary line is dashed because points on the line are not included. For y <= 2x + 1, the boundary line is solid because points on the line are included in the solution set.

Question 15

A graph shows the overlap of these inequalities: x > 0, y > 0, and y < x + 2. Describe the solution set in words.

Accepted Answer

The solution set is all points in the first quadrant that lie below the line y = x + 2. Because the inequalities are strict, the axes and the line are not included in the solution set.

Math: Systems of Inequalities

Math: Systems of Inequalities

More Math Worksheets

More Grade 9-12 Worksheets