Question 1

Which of the following is the correct solution set in interval notation for the inequality $$3x - 9x + 4 \u003c 4$$?

Accepted Answer

$$(-\frac{2}{3}, +\infty)$$

Question 2

If $$|x| \u003c 1$$, what is the solution set for $$x$$?

Accepted Answer

$$(-1, 1)$$

Question 3

Suppose a company produces two products, A and B. Each unit of A requires 2 hours of labor and 3 kg of material. Each unit of B requires 1 hour of labor and 4 kg of material. If the company has at most 40 hours of labor and 60 kg of material available per week, which system of inequalities represents the constraints?

Accepted Answer

$$2x + 1y \leq 40$$ and $$3x + 4y \leq 60$$

Question 4

Which of the following statements is TRUE about the solution to a system of two linear inequalities in two variables?

Accepted Answer

It is the region where the solution sets of both inequalities overlap.

Question 5

Given the absolute value inequality $$|x - 3| \u003e 1$$, what is the solution set?

Accepted Answer

$$(-\infty, 2) \cup (4, +\infty)$$

Question 6

A business wants to maximize its weekly profit $$P = 8x_1 + 5x_2$$, where $$x_1 is $$the number of Space Rays and $$x_2 is $$the number of Zappers produced. If each Space Ray requires 1 pound of plastic and 5 minutes of labor, and each Zapper requires 2 pounds of plastic and 4 minutes of labor, with 1000 pounds of plastic and 2000 minutes of labor available per week, which of the following is the correct set of constraints?

Accepted Answer

$$x_1 + 2x_2$$ \leq 1000$$ and 5x_1 + 4x_2$$ \leq 2000$$

Question 7

If $$5t + 8 \leq 7(1 + t)$$, what is the solution set for $$t$$?

Accepted Answer

$$[\frac{1}{2}, +\infty)$$

Question 8

Which of the following best describes the boundary line for the inequality $$y \u003e x + 4$$ when graphing in the $$xy-$$plane?

Accepted Answer

A dashed line at $$y = x + 4$$ not included in the solution set.

Question 9

Suppose you are solving the system: $$y \geq 2x - 1$$ and $$y \u003c x + 1. $$Which of the following points is in the solution region?

Accepted Answer

$$(-1, -1)$$

Question 10

A student wants to display at least 25 photos at a booth, with small photos costing $$4$$ each and large photos costing $$10$$ each. If the student cannot spend more than $$200$$, which system of inequalities models this situation (let $$x = $$number of small photos, $$y = $$number of large photos)?

Accepted Answer

$$x + y \geq 25$$ and $$4x + 10y \leq 200$$

Question 11

If $$|2x - 1| \leq 3$$, what is the solution set for $$x$$?

Accepted Answer

$$[-1, 2]$$

Question 12

Which of the following is a property of inequalities when multiplying both sides by a negative number?

Accepted Answer

The direction of the inequality sign reverses.

Question 13

A company must produce at least 4000 lbs of mineral A and 2000 lbs of mineral B. Ore 1 yields 2 lbs of A and 1 lb of B per ton, and Ore 2 yields 1 lb of A and 2 lbs of B per ton. Let $$x$$ and $$y be $$the tons of Ore 1 and Ore 2 processed, respectively. Which system of inequalities represents the production constraints?

Accepted Answer

$$2x + y \geq 4000$$ and $$x + 2y \geq 2000$$

Question 14

Which of the following is the correct interval notation for the solution to $$1 \u003c x \u003c 5$$?

Accepted Answer

$$(1, 5)$$

Question 15

In a linear programming problem, what is the objective function?

Accepted Answer

A function that is to be maximized or minimized, such as profit or cost.

Question 16

If $$|x| \geq 1$$, what is the solution set for $$x$$?

Accepted Answer

$$(-\infty, -1] \cup [1, +\infty)$$

Question 17

Which of the following is always true about the feasible region in a linear programming problem with two variables?

Accepted Answer

It is the intersection of the half-planes defined by the constraints.

Question 18

If a linear programming problem has an optimal solution, where will it be found?

Accepted Answer

At an extreme point (vertex) of the feasible region.

Inequalities and Linear Programming: Applied Mathematics for Business

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