Deck 24: Inequalities and Linear Programming

Question 1

Graph the following inequality as a ray or interval on the number line: $x>9$

Answer 1

Question 2

Graph the region in which both of the following two inequalities are satisfied: $y \geq x-8$ and $y \leq x / 2-6$

Answer 2

Question 3

Label each inequality as conditional, unconditional, linear, or nonlinear:
(a) $x^{2}+2>1$
(b) $x+4<2 x-3$
(c) $x^{2}-5 x+4>5$
(d) $5 x+1>x^{2}$

Answer 3

(a) unconditional, nonlinear
(b) conditional, linear
(c) conditional, nonlinear
(d) conditional, nonlinear

Question 4

Rewrite each double inequality as two separate inequalities:

Question 5

Graph the inequality on the number line: $x \leq-4$

Question 6

Graph the inequality: $y\geq 2 x-1$

Question 7

Graph the inequalities: $y \leq 3 x-4, y \geq 2 x+5$

Question 8

Graph the inequalities: $y<8-5 x^{2}, y\geq 4 x+8$

Question 9

Write $(-3,8)$ as an inequality.

Question 10

Write using bracket notation.

Question 11

Graph the inequality on the number line: $-2 \leq x<1$

Question 12

Write $-5 \leq x<7$ using bracket notation.

Question 13

Rewrite each double inequality as two separate inequalities:

(c) $-1 \leq x \leq 8$
(d) $-10 \leq x<-5$

Question 14

Rewrite each pair of inequalities as a double inequality:
(a) $x \geq 2$ and $x<15$
(b) $x>-4$ and $x \leq 25$
(c) $x \geq-5$ and $x \leq 12$
(d) $x>20$ and $x<64$

Question 15

Solve the inequality: $3 x-9<4 x+2$

Question 16

Solve the inequality:

Question 17

Solve the inequality: $|5 x|<15$

Question 18

Solve the inequality: $|4 x+3|>5$

Question 19

Solve the inequality: $4 x-8 \geq 3(4 x+1)+13$

Question 20

Solve the inequality: $\frac{x^{2}-3 x}{2}>\frac{-x^{2}+5 x+90}{2}$

Question 21

Solve the inequality: $x^{2}+x<6$

Question 22

Solve the inequality: $|5 x-3|<2 x$

Question 23

Solve the inequality: $\left|x^{2}-5 x+5\right| \leq 1$

Question 24

Solve the inequality: $|3-5 x|<2-3 x$

Question 25

Find the values of $x$ for which the expression is real:
$\overline{\sqrt{x^{3}+8 x^{2}-9 x}}$

Question 26

Find the values of $x$ for which the expression is real:
$\overline{\sqrt{\left(x^{2}-3 x+2\right)\left(x^{2}-x-2\right)}}$

Question 27

Solve the inequality: $2(4-x)+x \leq 5-2(1-2 x)$

Question 28

Solve the inequality: $2 x \leq 4 x+7<2 x+20$

Question 29

Solve the inequality: $2 x^{2}+x \leq 15$

Question 30

A company can make a toaster for $\$ 9.50$ and sell it for $\$ 29.75$ . Find the minimum number of toasters that must be sold for the company to make a profit if it has an overhead cost of $\$ 8100$ per week.

Question 31

Solve the inequality: $2 x-15 \geq 5+6 x$

Question 32

Solve the inequality: $|5 x-21| \geq 2 x$

Question 33

It costs a company $\$ 500$ in labor to make 75 parts, $\$ 200$ in material to make 30 parts, and there is an overhead cost of $\$ 24500$ a month. If each part is sold for $\$ 45$ , find the minimum number of parts that must be sold each month for the company to make a profit.

Question 34

Maximize the function $z=-2 x+4 y$ under the constraints $x+y\leq 2,8 x-y\geq-8$ , and $y\geq 0$ .

Question 35

Maximize the function $z=3 x+2 y$ under the constraints $y-x \leq 1, x+2 y \leq 5, x\geq 0$ , and $y \geq 0$ .

Question 36

Maximize the function $z=-x+2 y$ under the constraints $y \geq 4 x$ and $x+y \leq 10, x \geq 0, y \geq 0$ .

Question 37

Maximize the function $z=4 x-3 y$ under the constraints $2 x+y \leq 8, x-2 y \leq 5,4 x-y\geq-16, x\geq 0, y\geq 0$ .

Question 38

A company makes two models of televisions. Model A requires 3 hours to make and $\$ 125$ in materials and yields a profit of $\$ 175$ . Model B requires 7 hours to make and $\$ 200$ in materials and yields a profit of $\$ 350$ . There are 1010 hours and $\$ 34750$ available per week to make the televisions. How many of each television model must be made each month in order to maximize profit?

Question 39

A company makes two different laptops. Model A requires 4 hours to make and $\$ 350$ in materials and yields a profit of $\$ 750$ . Model B requires 6 hours to make and $\$ 400$ in materials and yields a profit of $\$ 900$ . There are 1000 hours and $\$ 70000$ available per week to make laptops. How many of each laptop model must be made each month in order to maximize profit?