Solving Equations and Inequalities

Name the property illustrated by each equation. 

$\left(2+14\right)+3=3+\left(2+14\right)$

Name the property illustrated by each equation. 

$\left(1\frac{2}{7}\right)\left(\frac{7}{9}\right)=1$

Name the property illustrated by each equation. 

$2\sqrt{3}+5\sqrt{3}=\left(2+5\right)\sqrt{3}$

Name the property illustrated by each equation. 

35. $ab=1ab$

Use the properties of real to answer each question

 If $m+n=m$, what is the value of $n$?

NUMBER THEORY

Use the properties of real to answer each question

37.  If$m+n=0$, what is the value of $n$? What is $n$ called with respect to $m$?

Use the properties of real to answer each question

 If $mn=1$, what is the value of $n$? What is $n$ called with respect to $m$?

Use the properties of real to answer each question

  If $mn=m$, what is the value of $n$?

Use the following information.

The Greek mathematician Pythagoras believed that all things could be described by numbers. By “number” he meant positive integers.

To what set of numbers was Pythagoras referring when he spoke of “numbers?” 

MATH HISTORY

Use the following information.

The Greek mathematician Pythagoras believed that all things could be described by numbers. By “number” he meant positive integers.

41. Use the formula $c=\sqrt{2s^2}$ to calculate the length of the hypotenuse c, or the longest side, of this right triangle using s, the length of one leg.

Use the following information.

The Greek mathematician Pythagoras believed that all things could be described by numbers. By “number” he meant positive integers.

Explain why Pythagoras could not find a “number” to describe the value of c.

Identify the additive inverse and multiplicative inverse for each number 

$-10$

Identify the additive inverse and multiplicative inverse for each number 

44. $2.5$

Identify the additive inverse and multiplicative inverse for each number 

$-0.125$

Identify the additive inverse and multiplicative inverse for each number 

$-\frac{5}{8}$

Identify the additive inverse and multiplicative inverse for each number 

47. $\frac{4}{3}$

Identify the additive inverse and multiplicative inverse for each number 

$-4\frac{3}{5}$

Simplify each expression

$7a+3b-4a-5b$

Simplify each expression

$3x+5y+7x-3y$

Simplify each expression

51. $3\left(15x-9y\right)+5\left(4y-x\right)$

Simplify each expression

$2\left(10m-7a\right)+3\left(8a-3m\right)$

Simplify each expression

53. $8\left(r+7t\right)-4\left(13t+5r\right)$

Simplify each expression

$4\left(14c-10d\right)-6\left(d+4c\right)$

Simplify each expression

 $4\left(0.2m-0.3n\right)-6\left(0.7m-0.5n\right)$

Simplify each expression

56. $7\left(0.2p+0.3q\right)+5\left(0.6p-q\right)$

Simplify each expression

$\frac{1}{4}\left(6+20y\right)-\frac{1}{2}\left(19-8y\right)$

Simplify each expression

 $\frac{1}{6}\left(3x+5y\right)+\frac{2}{3}\left(\frac{3}{5}x-6y\right)$

Determine whether each statement is true or false. If false, give a counterexample.

Every whole number is an integer

Determine whether each statement is true or false. If false, give a counterexample.

60. Every integer is a whole number.

Determine whether each statement is true or false. If false, give a counterexample.

Every real number is irrational.

Determine whether each statement is true or false. If false, give a counterexample.

Every integer is a rational number.

WORK

Use the information below and in the table.

Andrea works as a hostess in a restaurant and is paid every two weeks.

63. If Andrea earns $\$6.50$ an hour, illustrate the Distributive Property by writing two expressions representing Andrea’s pay last week.

Use the information below and in the table.

Andrea works as a hostess in a restaurant and is paid every two weeks.

Find the mean or average number of hours Andrea worked each day, to the nearest tenth of an hour. Then use this average to predict her pay for a two-week pay period. Her hourly wage is $6.5

Mitena is making two types of cookies. The first recipe calls for $2\frac{1}{4}$cups of flour, and the second calls for $1\frac{1}{8}$ cups of flour. If Mitena wants to make 3 batches of the first recipe and 2 batches of the second recipe, how many cups of flour will she need? Use the properties of real numbers to show how Mitena could compute this amount mentally. Justify each step.

For exercise 68 and 69, use the graph at the right.

Evaluate the expression from Exercise 68 using the Distributive Property.

BASKETBALL

Use the diagram of an NCAA basketball court below.

66. Illustrate the Distributive Property by writing two expressions for the area of the basketball court.

BASKETBALL

Use the diagram of an NCAA basketball court below.

Evaluate the expression from Exercise 66 using the Distributive Property. What is the area of an NCAA basketball court?

SCHOOL SHOPPING

For exercise 68 and 69, use the graph at the right.

Illustrate the Distributive Property by writing two expression to represent the amount that the average student spends shopping for school at specialty stores and department stores.

CRITICAL THINKING

Is the Distributive Property also true for division? In other words, does $\frac{b+c}{a}=\frac{b}{a}+\frac{c}{a},a\ne 0$? If so, give an example and explain why it is true. If not true, give a counterexample.

Answer the question that was posed at the beginning of the lesson.

How is the Distributive Property useful in calculating store savings?

Include the following in your answer:

• an explanation of how the Distributive Property could be used to calculate the coupon savings listed on a grocery receipt, and
• an example of how the Distributive Property could be used to calculate the savings from a clothing store sale where all items were discounted by the same percent.

If $a$ and $b$ are natural numbers, then which of the following must also be a natural number?

$\text{I. }a-b\text{ II. }ab\text{ III. }\frac{a}{b}$$\text{I. }a-b\text{ II. }ab\text{ III. }\frac{a}{b}$

(A) I only      (B) II only      (C) III only       (D) I and II only      (E) II and III only

If $x=1.4$, find the value of $27\left(x+1.2\right)-26\left(x+1.2\right)$ 

(A) 1      (B) $-0.4$      (C) $2.6$       (D) $65$

Find the value of each expression.

$-4+5\left(7-2^3\right)$ 

Find the value of each expression.

$\frac{18+3\times 4}{13-8}$

4. Evaluate $a^3+b\left(9-c\right)$ if  $a=-2,b=\frac{1}{3},$ and $c=-12$

Find the amount of current I (in amperes) produced if the electromotive force E is 2.5 volts, the circuit resistance R is 1.05 ohms, and the resistance r within a battery is 0.2 ohm. Use the formula $I=\frac{E}{R+r}$

Name the sets of numbers to which each number belongs.

 3.5

Name the sets of numbers to which each number belongs.

$\sqrt{100}$

8. Name the property illustrated by $bc+\left(-bc\right)=0$

9. Name the additive inverse and multiplicative inverse $\frac{6}{7}$