Expressions, Equations, and Functions

Example 2 Identify the hypothesis and conclusion of each statement.

Then write each statement in if-then form.

August has 31 days.

Identify the hypothesis and conclusion of each statement.

Then write each statement in if-then form.

Science teachers like to conduct experiments.

Example 3 Determine whether a valid conclusion follows from the statement below for each given condition. If a valid conclusion does not follow, write no valid conclusion and explain why.

If Belinda scores higher than $90\%$ on the exam, then she will receive an A for the course.

Belinda scores a $91\%$ on the exam.

Example 3 Determine whether a valid conclusion follows from the statement below for each given condition. If a valid conclusion does not follow, write no valid conclusion and explain why.

If Belinda scores higher than $90\%$ on the exam, then she will receive an A for the course.

Belinda scores an $89\%$ on the exam.

Determine whether a valid conclusion follows from the statement below for each given condition. If a valid conclusion does not follow, write no valid conclusion and explain why.

If Belinda scores higher than $90\%$ on the exam, then she will receive an A for the course.

Belinda receives an A for the course.

Example 3 Determine whether a valid conclusion follows from the statement below for each given condition. If a valid conclusion does not follow, write no valid conclusion and explain why.

If Belinda scores higher than $90\%$ on the exam, then she will receive an A for the course.

Belinda receives a B for the course.

Example 4 Find a counterexample for each conditional statement.

If you live in London, then you live in England.

Find a counterexample for each conditional statement.

If you attend the banquet, then you will eat the food.

Example 4 Find a counterexample for each conditional statement.

If the four sides of a quadrilateral are congruent, then the shape is a square.

Find a counterexample for each conditional statement.

If a number is divisible by 3, then the number is odd.

Example 4 Find a counterexample for each conditional statement.

If $3x+17\le 53$ then $x<12$.

Example 4 Find a counterexample for each conditional statement.

If $x^2=1$ then x must equal 1.

Find a counterexample for each conditional statement.

If an animal has spots, then it is a Dalmatian.

Example 4 Find a counterexample for each conditional statement.

If a number is prime, then it is an odd number.

Find a counterexample for each conditional statement.

If an animal cannot fly, then the animal is not a bird.

Determine whether a valid conclusion follows from the statement below for each given condition. If a valid conclusion does not follow, write no valid conclusion and explain why.

If the dimensions of rectangle $ABCD$ are doubled, then the perimeter is doubled.

a. The new rectangle measures 16 inches by 10 inches.

b. The perimeter of the new rectangle is 52 inches.

GEOMETRY Use the following statement. 

If there are three line-segments $\overline{AB},\overline{BC}$ and  $\overline{CD}$ then they form a triangle.

a. Draw a diagram to provide an example for the conditional statement.

b. Draw a diagram to provide a counterexample for the conditional statement.

GROUNDHOG DAY On Groundhog Day, some people say that if a groundhog sees its shadow, then there will be 6 more weeks of winter. If it does not see its shadow, then there will be an early spring.

a. The most famous groundhog, Punxsutawney Phil in Pennsylvania, sees his shadow 85% of the time. Write an algebraic expression to represent how many times he sees his shadow in y years.

b. The table lists each possible scenario. From the given conditional statement, determine whether this is true or false.

c. Of the situations listed in the table, explain which situation could be considered a counterexample to the original statement.

CHALLENGE Determine whether the following statement is always true. If not, provide a counterexample.

If $\mathbf{2}\left(\mathbf{b}+\mathbf{c}\right)=\mathbf{2}\mathbf{b}+\mathbf{2}\mathbf{c}$, then $\mathbf{2}+\left(\mathbf{b}\cdot \mathbf{c}\right)=\left(\mathbf{2}+\mathbf{b}\right)\left(\mathbf{2}+\mathbf{c}\right)$.

For what values of n is the opposite of n greater than n? For what values of n is the opposite of n less than n? For what values is n equal to its opposite?

OPEN-ENDED Write a conditional statement. Label the hypothesis and conclusion.

Determine whether this statement is true or false. If the length of a rectangle is doubled, then the area of the rectangle is doubled. Justify your answer.

OPEN-ENDED Write a conditional statement. Write a counterexample to the statement. Explain your reasoning.

Explain how deductive reasoning is used to show that a conditional is true or false.

Which value of $b$ serves as a counterexample to the statement $\mathbf{2}\mathbf{b}<\mathbf{3}\mathbf{b}$?

A. $-\mathbf{4}$     C. $\frac{1}{2}$

B. $\frac{1}{4}$      D. 4

SHORT RESPONSE A deli serves boxed lunches with a sandwich, fruit, and a dessert. The sandwich choices are turkey, roast beef, or ham. The fruit choices are an orange or an apple. The dessert choices are a cookie or a brownie. How many different boxed lunches does the deli serve?

Which illustrates the Transitive Property of Equality?

F If $\mathbf{c}=\mathbf{1}$ then $\mathbf{c}\cdot \frac{1}{c}=\mathbf{1}$.

G If $\mathbf{c}=\mathbf{d}$ and $\mathbf{d}=\mathbf{f}$ then $\mathbf{c}=\mathbf{f}$.

H If $\mathbf{c}=\mathbf{d}$ then $\mathbf{d}=\mathbf{c}$.

J If $\mathbf{c}=\mathbf{d}$ and $\mathbf{d}=\mathbf{c}$ then $\mathbf{c}=\mathbf{1}$.

Simplify the expression $\mathbf{5}\mathbf{d}\left(\mathbf{7}-\mathbf{3}\right)-\mathbf{16}\mathbf{d}+\mathbf{3}\cdot \mathbf{2}\mathbf{d}$.

A 10 d                        C 21 d

В 14 d                        D 25 d

Determine whether each relation is a function. (Lesson 1-7)

Determine whether each relation is a function. (Lesson 1-7)

$\left\{\left(0,2\right),\left(3,5\right),\left(0,-1\right),\left(-2,4\right)\right\}$

Determine whether each relation is a function. (Lesson 1-7)

GEOMETRY Express the relation in the graph at the right as a set of ordered pairs and describe the domain and range. (Lesson 1-6)

Robert has 30 socks in his sock drawer. 16 of the socks are white, 6 are black, 2 are red, and 6 are yellow. What is the probability that he randomly pulls out a black sock?

Find the perimeter of each figure. (Lesson 0-7)

Find the perimeter of each figure. (Lesson 0-7)

Evaluate each expression.

$7^2$

Evaluate each expression. (Lesson 1-2)

${\left(-9\right)}^2$

Evaluate each expression. (Lesson 1-2)

$2.7^2$

Evaluate each expression.

${\left(-12.25\right)}^2$

Evaluate each expression. (Lesson 1-2)

$5^2$

Evaluate each expression. (Lesson 1-2)

${25}^2$

State whether each sentence is true or false, replace the underlined term to make a true sentence.

A coordinate system is formed by intersecting number lines

State whether each sentence is true or false. If false, replace the underlined term to make a true sentence.

An exponent indicates the number of times the base is to be used as a factor.

State whether each sentence is true or false; replace the underlined term to make a true sentence.

1. An expression is in simplest form when it contains like terms and parentheses

State whether each sentence is true or false; replace the underlined term to make a true sentence.

In an expression involving multiplication, the quantities being multiplied are called factors.

State whether each sentence is true or false. If false, replace the underlined term to make a true sentence.

In a function, there is exactly one output for each input.

State whether each sentence is true or false. If false, replace the underlined term to make a true sentence.

Order of operations tells us to perform multiplication before subtraction.

State whether each sentence is true or false. If false, replace the underlined term to make a true sentence.

Since the product of any number and 1 is equal to the number, 1 is called the multiplicative inverse.

Write a verbal expression for each algebraic expression.

$3x^2$

Write a verbal expression for each algebraic expression.

$5+6m^3$