Q: If a company has $$3,900$$ minutes of assembly and $$3,300$$ minutes of painting available each week, and the basic product requires $$20$$ minutes of assembly and $$15$$ minutes of painting, while the premium product requires $$30$$ minutes of assembly and $$30$$ minutes of painting, which system of inequalities models the production limits?

$$\begin{cases} 20b + 30p \leq 3900 \\ 15b + 30p \leq 3300 \end{cases}$$

Q: A person's diet consists of fish and mushrooms. Each serving of fish is $$500$$ calories and $$40$$ grams of protein. Each serving of mushrooms is $$50$$ calories and $$1$$ gram of protein. If the person requires at least $$1,000$$ calories and at least $$20$$ grams of protein, which system of inequalities models this situation?

$$\begin{cases} 500f + 50m \geq 1000 \\ 40f + m \geq 20 \end{cases}$$

Q: A company can produce up to $$195$$ basic products and $$0$$ premium products, or $$120$$ basic and $$50$$ premium products. If basic products sell for $$30$$ and premium for $$40$$, which production plan yields the highest revenue?

$$195$$ basic and $$0$$ premium products: $$30 \times 195 + 40 \times 0 = 5,850$$

Q: A person's diet consists of fish and mushrooms. Each serving of fish is $$500$$ calories and $$40$$ grams of protein. Each serving of mushrooms is $$50$$ calories and $$1$$ gram of protein. If the person wants less than $$1,000$$ calories and at least $$20$$ grams of protein, which system of inequalities models this situation?

$$\begin{cases} 500f + 50m \u003c 1000 \\ 40f + m \geq 20 \end{cases}$$

Question 1

Which graph best represents the solution set to the inequality $$y \u003c 2x + 1$$?

Accepted Answer

A dashed line for $$y = 2x + 1$$ with shading below the line.

Question 2

Given the system of inequalities $$y \leq 2x + 1$$ and $$y \geq -x$$, which region on the graph represents the solution set?

Accepted Answer

The region where the shaded areas for both inequalities overlap.

Question 3

A store sells peanuts for $$4/pound$$ and cashews for $$8/pound. If $$the mix must cost $$6 or $$less per pound, which inequality models the possible combinations of peanuts ($$p) $$and cashews ($$c) in $$one pound of mix?

Accepted Answer

$$4p + 8c \leq 6$$

Question 4

If a company has $$3,900$$ minutes of assembly and $$3,300$$ minutes of painting available each week, and the basic product requires $$20$$ minutes of assembly and $$15$$ minutes of painting, while the premium product requires $$30$$ minutes of assembly and $$30$$ minutes of painting, which system of inequalities models the production limits?

Accepted Answer

$$\begin{cases} 20b + 30p \leq 3900 \ 15b + 30p \leq 3300 \end{cases}$$

Question 5

When graphing the inequality $$y \geq 2x - 1$$, which of the following is correct?

Accepted Answer

Draw a solid line for $$y = 2x - 1$$ and shade above the line.

Question 6

A person's diet consists of fish and mushrooms. Each serving of fish is $$500$$ calories and $$40$$ grams of protein. Each serving of mushrooms is $$50$$ calories and $$1$$ gram of protein. If the person requires at least $$1,000$$ calories and at least $$20$$ grams of protein, which system of inequalities models this situation?

Accepted Answer

$$\begin{cases} 500f + 50m \geq 1000 \ 40f + m \geq 20 \end{cases}$$

Question 7

If you graph $$y \u003c 2x + 1$$, which test point would be easiest to use to determine which side of the line to shade?

Accepted Answer

$$(3,0)$$

Question 8

When solving $$2x + 4 \geq y$$, what is the equivalent inequality in slope-intercept form?

Accepted Answer

$$y \leq 2x + 4$$

Question 9

Which of the following best describes the solution set to a system of linear inequalities?

Accepted Answer

The region where the shaded areas of all inequalities overlap.

Question 10

A company can produce up to $$195$$ basic products and $$0$$ premium products, or $$120$$ basic and $$50$$ premium products. If basic products sell for $$30$$ and premium for $$40$$, which production plan yields the highest revenue?

Accepted Answer

$$195$$ basic and $$0$$ premium products: $$30 	imes 195 + 40 	imes 0 = 5,850$$

Question 11

If the solution set to a linear inequality is a half-plane, what determines whether the boundary line is solid or dashed?

Accepted Answer

Solid for $$\leq or$$ $$\geq$$, dashed for $$.

Question 12

Which of the following is a real-world application of systems of linear inequalities?

Accepted Answer

Determining possible combinations of products a company can produce given staffing constraints.

Question 13

If you want to graph $$2x - y \u003c 4$$, what is the first step to rewrite the inequality for graphing?

Accepted Answer

$$y \u003e 2x - 4$$

Question 14

A person's diet consists of fish and mushrooms. Each serving of fish is $$500$$ calories and $$40$$ grams of protein. Each serving of mushrooms is $$50$$ calories and $$1$$ gram of protein. If the person wants less than $$1,000$$ calories and at least $$20$$ grams of protein, which system of inequalities models this situation?

Accepted Answer

$$\begin{cases} 500f + 50m \u003c 1000 \ 40f + m \geq 20 \end{cases}$$

Question 15

If you graph $$y \leq -x + 2$$, which side of the line should be shaded?

Accepted Answer

Below the line.

Question 16

Which step is necessary when dividing both sides of an inequality by a negative number?

Accepted Answer

Reverse the direction of the inequality sign.

Question 17

If the solution set to a system of inequalities is restricted to the first quadrant, what does this mean?

Accepted Answer

Only non-negative values for both variables are considered.

Question 18

Which of the following is NOT a step in graphing a linear inequality?

Accepted Answer

Find the roots of the equation.

Question 19

If a company produces $$b$$ basic and $$p$$ premium products, and $$b \geq 0$$, $$p \geq 0$$, what does this restriction represent?

Accepted Answer

The company cannot produce negative quantities of products.

Section 5.3: Linear Inequalities and Systems of Linear Inequalities

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