Solving Equations and Inequalities

Name the sets of numbers to which each number belongs.

$\sqrt{\mathbf{64}}$

Name the property illustrated by each equation or statement.

$\left(\mathbf{7}·\mathbf{s}\right)·\mathbf{t}\mathbf{=}\mathbf{7}·\left(\mathbf{s}·\mathbf{t}\right)$

Name the property illustrated by each equation or statement.

14. If $\left(\mathbf{r}\mathbf{+}\mathbf{s}\right)\mathbf{t}\mathbf{=}\mathbf{rt}\mathbf{+}\mathbf{st}$, then $\mathbf{rt}\mathbf{+}\mathbf{st}\mathbf{=}\left(\mathbf{r}\mathbf{+}\mathbf{s}\right)\mathbf{t}$.

Name the property illustrated by each equation or statement.

$\left(\mathbf{3}·\frac{\mathbf{1}}{\mathbf{3}}\right)·\mathbf{7}\mathbf{=}\left(\mathbf{3}·\frac{\mathbf{1}}{\mathbf{3}}\right)·\mathbf{7}$

Name the property illustrated by each equation or statement.

 $\left(\mathbf{6}\mathbf{-}\mathbf{2}\right)\mathbf{a}\mathbf{-}\mathbf{3}\mathbf{b}\mathbf{=}\mathbf{4}\mathbf{a}\mathbf{-}\mathbf{3}\mathbf{b}$

Name the property illustrated by each equation or statement.

 $\left(\mathbf{4}\mathbf{+}\mathbf{x}\right)\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{y}\mathbf{+}\left(\mathbf{4}\mathbf{+}\mathbf{x}\right)$

Name the property illustrated by each equation or statement.

If $\mathbf{5}\left(\mathbf{3}\right)\mathbf{+}\mathbf{7}\mathbf{=}\mathbf{15}\mathbf{+}\mathbf{7}$ and $\mathbf{15}\mathbf{+}\mathbf{7}\mathbf{=}\mathbf{22}$, then $\mathbf{5}\left(\mathbf{3}\right)\mathbf{+}\mathbf{7}\mathbf{=}\mathbf{22}$.

Solve each equation. Check your solution.

 $\mathbf{5}\mathbf{t}\mathbf{-}\mathbf{3}\mathbf{=}\mathbf{-}\mathbf{2}\mathbf{t}\mathbf{+}\mathbf{10}$

Solve each equation. Check your solution.

$\mathbf{2}\mathbf{x}\mathbf{-}\mathbf{7}\mathbf{-}\left(x-5\right)\mathbf{=}\mathbf{0}$

Solve each equation. Check your solutions.

$\mathbf{5}\mathbf{m}\mathbf{-}\left(\mathbf{5}\mathbf{+}\mathbf{4}\mathbf{m}\right)\mathbf{=}\left(\mathbf{3}\mathbf{+}\mathbf{m}\right)\mathbf{-}\mathbf{8}$ 

Solve each equation. Check your solution. 

$\left|\mathbf{8}\mathbf{\omega }\mathbf{+}\mathbf{2}\right|\mathbf{+}\mathbf{2}\mathbf{=}\mathbf{0}$

Solve each equation. Check your solution.

23. $\mathbf{12}\left|\frac{\mathbf{1}}{\mathbf{2}}\mathbf{y}\mathbf{+}\mathbf{3}\right|\mathbf{=}\mathbf{6}$

Solve each equation. Check your solution.

$\mathbf{2}\left|\mathbf{2}\mathbf{y}\mathbf{-}\mathbf{6}\right|\mathbf{+}\mathbf{4}\mathbf{=}\mathbf{8}$

Solve each inequality. Describe the solution set using set builder or interval notation. Then graph the solution set on a number line.

$\mathbf{4}>\mathbf{b}\mathbf{+}\mathbf{1}$

Solve each inequality. Describe the solution set using set builder or interval notation. Then graph the solution set on a number line.

 $\mathbf{3}\mathbf{q}\mathbf{+}\mathbf{7}\ge \mathbf{13}$

Solve each inequality. Describe the solution set using set builder or interval notation. Then graph the solution set on a number line.

 $\mathbf{5}\left(\mathbf{3}\mathbf{x}\mathbf{-}\mathbf{5}\right)\mathbf{+}\mathbf{x}<\mathbf{2}\left(\mathbf{4}\mathbf{x}\mathbf{-}\mathbf{1}\right)\mathbf{+}\mathbf{1}$

Solve each inequality. Describe the solution set using set builder or interval notation. Then graph the solution set on a number line.

 $\left|\mathbf{5}\mathbf{+}\mathbf{k}\right|\le \mathbf{8}$

Solve each inequality. Describe the solution set using set builder or interval notation. Then graph the solution set on a number line.

 $\mathbf{-}\mathbf{12}<\mathbf{7}\mathbf{d}\mathbf{-}\mathbf{5}\le \mathbf{9}$

Solve each inequality. Describe the solution set using set builder or interval notation. Then graph the solution set on a number line.

 $\left|\mathbf{3}\mathbf{y}\mathbf{-}\mathbf{1}\right|>\mathbf{5}$

For exercises 31 and 32, define a variable, write an equation or inequality, and solve the problem.

31. Mrs. Denney is renting a car that gets 35 miles per gallon. The rental charge is $\mathbf{\backslash (}\mathbf{19}\mathbf{.}\mathbf{50}$ a day plus $\mathbf{18}\mathbf{\backslash }\mathbf{notc}$ per mile. Her company will reimburse her for $\mathbf{\backslash )}\mathbf{33}$ of this portion of her travel expenses. If Mrs. Denney rents the car for 1 day, find the maximum number of miles that will be paid for by her company.

For exercises 31 and 32, define a variable, write an equation or inequality, and solve the problem.

32. To receive a B in his English class, Nick must have an average score of at least 80 on five tests. He scored 87,89,76 and 77 on his first four tests. What must he score on the last test to receive a B in the class?

If $\frac{\mathbf{a}}{\mathbf{b}}\mathbf{=}\mathbf{8}$ and $\mathbf{ac}\mathbf{-}\mathbf{5}\mathbf{=}\mathbf{11}$, then $\mathbf{bc}\mathbf{=}$

A.93        B. 2      C. $\frac{\mathbf{5}}{\mathbf{8}}$     D. cannot be determined

Question: 

1. In the square at the right, state the value of $\text{x}$.

A. 1                             B. 2

C. 3                            D. 4

On a college math test, 18 students earned an A. This number is exactly $30\%$ of the total number of students in the class. State how many students are in the class.

A. $5$                                  B. 23

C. $48$                               D. 60

A student computed the average of her 7 test scores by adding the scores and dividing this total by the number of tests. The average was 87. On her next test, she scored 79. State what is her new test average.

A. 83                                  B. 84

C. 85                                  D. 86

4. If the perimeter of $\Delta \text{PQR}$ is 3 times the length of $\text{PQ}$, then $\text{PR}=\_\_\_\_$.

A. 4                                  B. 6

C. 7                                  D. 8

5. If a different number is selected from each of the three sets shown below, what is the greatest sum these 3 numbers could have. 

$\text{R}=\left\{3,6,7\right\}$; $\text{S}=\left\{2,4,7\right\}$; $\text{T}=\left\{1,3,7\right\}$.

A. 13                                 B. 14

C. 17                                   D. 21

A pitcher contains $\text{a}$ ounces of orange juice. If $\text{b}$ ounces of juice are poured from pitcher into each of $\text{c}$ glasses, state which expression represents the amount of remaining in the pitcher.

A. $\frac{\text{a}}{\text{b}}+\text{c}$                                 B. $\text{ab}-\text{c}$

C. $\text{a}-\text{bc}$                                 D. $\frac{\text{a}}{\text{bc}}$

The sum of three consecutive integers is 135. State what is the greatest of the three integers.

A. 43                                 B. 44

C. 45                                   D. 46

The ratio of girls to boys in a class is 5 to 4. If there are total of 27 students in the class, state how many are girls.

A. 15                                 B. 12

C. 9                           D. 5

9. State for which of the following ordered pairs $\left(\text{x},\text{y}\right)$ is $\text{x}+\text{y}>3$ and $\text{x}-\text{y}<-2$.

A. $\left(0,3\right)$                                  B. $\left(3,4\right)$

C. $\left(5,3\right)$                                 D. $\left(2,5\right)$

 If the area of $\Delta \text{ABD}$ is 280, state the area of the polygon $\text{ABCD}$.

A. $560$                                 B. $630$

C. $700$                                   D. $840$

In the triangle below, $\mathbf{x}$ and $\mathbf{y}$ are integers. If $\mathbf{25}\mathbf{<}\mathbf{y}\mathbf{<}\mathbf{30}$, what is one possible value of $\mathbf{x}$?

If $\mathbf{n}$ and $\mathbf{p}$ are each different positive integers and $\mathbf{n}\mathbf{+}\mathbf{p}\mathbf{=}\mathbf{4}$, what is one possible value of $\mathbf{3}\mathbf{n}\mathbf{+}\mathbf{4}\mathbf{p}$?

In the figure at the right, state the value of $\text{x}$.

One half quart of lemonade concentrate is mixed with $\mathbf{1}\frac{\mathbf{1}}{\mathbf{2}}$ quarts of water to make lemonade for 6 people. If you use the same proportions of concentrate and water, how many quarts of lemonade concentrate are needed to make lemonade for 21 people? 

15. If 25 percent of 300 is equal to 500 percent of $\text{t}$, then state $\text{t}$ is equal to what number.

In the figure below, what is the area of the shaded square in square units?

There are 140 students in the school band. One of these students will be selected at random to be the student representative. If the probability that a brass player is selected is $\frac{\mathbf{2}}{\mathbf{5}}$, how many brass players are in the band?

A shelf holds fewer than 50 cans. If all of the cans on this shelf were put into stacks of five cans each, no cans would remain. If the same cans were put into stacks of three cans each, one can would remain. What is the greatest number of cans that could be on the shelf?

Compare the quantity in Column A and the quantity in Column B. Then determine whether:

A. the quantity in Column A is greater;

B. the quantity in Column B is greater;

C. the two quantities are equal;

D. the relationship cannot be determined from the information given.

Column A     Column B

$\frac{\frac{3}{4}}{{\left(\frac{3}{4}\right)}^2}$                 $\frac{4}{3}$

Compare the quantity in Column A and the quantity in Column B. Then determine whether:

A. the quantity in Column A is greater;

B. the quantity in Column B is greater;

C. the two quantities are equal;

D. the relationship cannot be determined from the information given.

Column A     Column B

$\text{x}+13$                 $\text{x}+14$

Compare the quantity in Column A and the quantity in Column B. Then determine whether:

A. the quantity in Column A is greater;

B. the quantity in Column B is greater;

C. the two quantities are equal;

D. the relationship cannot be determined from the information given.

$0<\text{s}<\frac{3}{4}$

Column A     Column B

      $1$                   $3\text{s}$

Compare the quantity in Column A and the quantity in Column B. Then determine whether:

A. The quantity in Column A is greater.

B. The quantity in Column B is greater.

C. The two quantities are equal.

D. The relationship cannot be determined from the information given.

Compare the quantity in Column A and the quantity in Column B. Then determine whether:

A. the quantity in Column A is greater;

B. the quantity in Column B is greater;

C. the two quantities are equal;

D. the relationship cannot be determined from the information given.

23. 

The average (arithmetic mean) of $\text{s}$ and $\text{t}$ is greater than the average of $\text{s}$ and $\text{w}$  

Column A     Column B

     $\text{w}$                   $\text{t}$