## Engage NY Eureka Math Algebra 1 Module 1 Lesson 15 Answer Key

### Eureka Math Algebra 1 Module 1 Lesson 15 Example Answer Key

Example 1.

Solve each system of equations and inequalities.

a. x+8=3 or x-6=2

Answer:

x=-5 or x=8

{-5,8}

b. 4x-9=0 or 3x+5=2

Answer:

x=\(\frac{9}{4}\) or x=-1

{-1,\(\frac{9}{4}\)}

c. x-6=1 and x+2=9

Answer:

x=7 and x=7

{7}

d. 2w-8=10 and w>9

Answer:

The empty set

Example 2.

Graph each compound sentence on a number line.

a. x=2 or x>6

Answer:

b. x≤-5 or x≥2

Answer:

Rewrite as a compound sentence, and graph the sentence on a number line.

c. 1≤x≤3

Answer:

x≥1 and x≤3

### Eureka Math Algebra 1 Module 1 Lesson 15 Exercise Answer Key

Exercise 1.

Determine whether each claim given below is true or false.

a. Right now, I am in math class and English class.

Answer:

False

b. Right now, I am in math class or English class.

Answer;

True (assuming they are answering this in class)

c. 3 + 5 = 8 and 5 < 7-1

Answer:

True

d. 10 + 2 ≠ 12 and 8-3 > 0

Answer:

False

e. 3 < 5 + 4 or 6 + 4 = 9

Answer:

True

f. 16-20 > 1 or 5.5 + 4.5 = 11

Answer:

False

These are all examples of declarative compound sentences.

g. When the two declarations in the sentences above were separated by “and,” what had to be true to make the statement true?

Answer:

Both declarations had to be true.

h. When the two declarations in the sentences above were separated by “or,” what had to be true to make the statement true?

Answer:

At least one declaration had to be true.

Discuss the following points with students:

→ The word “and” means the same thing in a compound mathematical sentence as it does in an English sentence.

→ If two clauses are separated by “and,” both clauses must be true for the entire compound statement to be deemed true.

→ The word “or” also means a similar thing in a compound mathematical sentence as it does in an English sentence. However, there is an important distinction: In English, the word “or” is commonly interpreted as the exclusive or, one condition or the other is true, but not both. In mathematics, either or both could be true.

→ If two clauses are separated by “or,” one or both of the clauses must be true for the entire compound statement to be deemed true.

Exploratory Challenge

Provide students with colored pencils, and allow them a couple of minutes to complete parts (a) through (c). Then, stop and discuss the results.

Exercise 2.

a. Using a colored pencil, graph the inequality x<3 on the number line below part (c). b. Using a different colored pencil, graph the inequality x>-1 on the same number line.

c. Using a third colored pencil, darken the section of the number line where x<3 and x>-1.

→ In order for the compound sentence x>-1 and x<3 to be true, what has to be true about x?

→ x has to be both greater than -1 and less than 3. (Students might also verbalize that it must be between -1 and 3, not including the points -1 and 3.)

→ On the graph, where do the solutions lie?

→ Between -1 and 3, not including the points -1 and 3.

Have students list some of the solutions to the compound inequality. Make sure to include examples of integer and non-integer solutions.

→ How many solutions are there to this compound inequality?

→ An infinite number.

Introduce the abbreviated way of writing this sentence:

→ Sometimes this is written as -1<x<3.

Use this notation to further illustrate the idea of x representing all numbers strictly between -1 and 3.

Allow students a couple of minutes to complete parts (d) through (f). Then, stop and discuss the results.

d. Using a colored pencil, graph the inequality x<-4 on the number line below part (f). e. Using a different colored pencil, graph the inequality x>0 on the same number line.

f. Using a third colored pencil, darken the section of the number line where x<-4 or x>0.

→ In order for the compound sentence x<-4 or x>0 to be true, what has to be true about x?

→ It could either be less than -4, or it could be greater than 0.

→ On the graph, where do the solutions lie?

→ To the left of -4 and to the right of 0.

Have students list solutions to the compound inequality. Make sure to include examples of integer and non-integer solutions.

→ How many solutions are there to this compound inequality?

→ Infinitely many.

→ Would it be acceptable to abbreviate this compound sentence as follows: 0<x<-4?

→ No.

→ Explain why not.

→ Those symbols suggest that x must be greater than zero and less than -4 at the same time, but the solution is calling for x to be either less than -4 or greater than zero.

Allow students a couple of minutes to complete parts (g) through (i), and discuss answers.

g. Graph the compound sentence x>-2 or x=-2 on the number line below.

Answer:

h. How could we abbreviate the sentence x>-2 or x=-2?

Answer:

x≥-2

i. Rewrite x≤4 as a compound sentence, and graph the solutions to the sentence on the number line below.

Answer:

x<4 or x=4

Exercise 3.

Consider the following two scenarios. For each, specify the variable and say, “W is the width of the rectangle,” for example, and write a compound inequality that represents the scenario given. Draw its solution set on a number line.

Answer:

Exercise 4.

Determine if each sentence is true or false. Explain your reasoning.

a. 8+6≤14 and \(\frac{1}{3}\)<\(\frac{1}{2}\)

Answer:

True

b. 5-8<0 or 10+13≠23

Answer:

True

Solve each system, and graph the solution on a number line.

c. x-9=0 or x+15=0

Answer:

{9,-15}

d. 5x-8=-23 or x+1=- 10

Answer:

{-3,-11}

Graph the solution set to each compound inequality on a number line.

e. x<-8 or x>-8

Answer:

f. 0<x≤10

Answer:

Write a compound inequality for each graph.

g.

Answer:

-3≤x≤4

h.

Answer:

x<-4 or x>0

i. A poll shows that a candidate is projected to receive 57% of the votes. If the margin for error is plus or minus 3%, write a compound inequality for the percentage of votes the candidate can expect to get.

Answer:

Let x= percentage of votes. 54≤x≤60

j. Mercury is one of only two elements that are liquid at room temperature. Mercury is nonliquid for temperatures less than -38.0° or greater than 673.8. Write a compound inequality for the temperatures at which mercury is nonliquid.

Answer:

Let x = temperatures (in degrees Fahrenheit) for which mercury is nonliquid. x<-38 or x>673.8

As an extension, students can come up with ways to alter parts (a) and (b) to make them false compound statements. Share several responses.

Ask the following:

→ What would be a more concise way of writing the sentence for part (e)?

→ x≠8.

→ For part (f), list some numbers that are solutions to the inequality.

→ What is the largest possible value of x?

→ 10.

→ What is the smallest possible value of x?

→ This is tougher to answer. x can be infinitely close to 0 but cannot equal zero. Therefore, there is no absolute smallest value for x in this case.

For parts (i) and (j), make sure students specify what the variable they choose represents.

Lesson Summary

In mathematical sentences, like in English sentences, a compound sentence separated by

AND is true if __both clauses are true__.

OR is true if __at least one of the clauses is true__.

### Eureka Math Algebra 1 Module 1 Lesson 15 Exit Ticket Answer Key

Question 1.

a. Solve the system, and graph the solution set on a number line.

x-15=5 or 2x+5=1

Answer:

x=20 or x=-2 {-2,20}

b. Write a different system of equations that would have the same solution set.

Answer:

Sample answer: x-20=0 or x+2=0

Question 2.

Swimming pools must have a certain amount of chlorine content. The United States standard for safe levels of chlorine in swimming pools is at least 1 part per million and no greater than 3 parts per million. Write a compound inequality for the acceptable range of chlorine levels.

Answer:

Let x= chlorine level in a swimming pool (in parts per million). 1≤x≤3

Question 3.

Consider each of the following compound sentences:

x<1 and x>-1 x<1 or x>-1

Does the change of the word from “and” to “or” change the solution set?

Use number line graphs to support your answer.

Answer:

For the first sentence, both statements must be true, so x can only equal values that are both greater than -1 and less than 1. For the second sentence, only one statement must be true, so x must be greater than -1 or less than 1. This means x can equal any number on the number line.

### Eureka Math Algebra 1 Module 1 Lesson 15 Problem Set Answer Key

Question 1.

Consider the inequality 0<x<3. a. Rewrite the inequality as a compound sentence. Answer; x>0 and x<3

b. Graph the inequality on a number line.

Answer:

c. How many solutions are there to the inequality? Explain.

Answer:

There are an infinite number of solutions. x can be any value between 0 and 3, which includes the integer values of 1 and 2 as well as non-integer values. The set of numbers between 0 and 3 is infinite.

d. What are the largest and smallest possible values for x? Explain.

Answer:

There is no absolute largest or absolute smallest value for x. x can be infinitely close to 0 or to 3 but cannot equal either value.

e. If the inequality is changed to 0≤x≤3, then what are the largest and smallest possible values for x?

Answer:

In this case, we can define the absolute maximum value to be 3 and the absolute minimum value to be 0.

Write a compound inequality for each graph.

Question 2.

Answer:

x<1 or x≥3

Question 3.

Answer:

x<2 or x>2, which can be written as x≠2

Write a single or compound inequality for each scenario.

Question 4.

The scores on the last test ranged from 65% to 100%.

Answer:

x= scores on last test 65≤x≤100

Question 5.

To ride the roller coaster, one must be at least 4 feet tall.

Answer:

x= height (in feet) to ride the roller coaster x≥4

Question 6.

Unsafe body temperatures are those lower than 96°F or above 104°F.

Answer:

x= body temperature (in degrees Fahrenheit) that are unsafe x<96 or x>104

Graph the solution(s) to each of the following on a number line.

Question 7.

x-4=0 and 3x+6=18

Answer:

Question 8.

x<5 and x≠0

Answer:

Question 9.

x≤-8 or x≥-1

Answer:

Question 10.

3(x-6)=3 or 5-x=2

Answer:

Question 11.

x<9 and x>7

Answer:

Question 12.

x+5<7 or x=2

Answer: