Into Math Grade 8 Module 3 Review Answer Key

We included HMH Into Math Grade 8 Answer Key PDF Module 3 Review to make students experts in learning maths.

HMH Into Math Grade 8 Module 3 Review Answer Key

Vocabulary

Use the equation in the box to answer Problems 1-2.

Question 1.
Circle each coefficient.
Answer:

A coefficient refers to a number or quantity placed with a variable. It is usually an integer that is multiplied by the variable and written next to it. The variables which do not have a number with them are assumed to be having 1 as their coefficient.
In the given expression, there are three terms. In the term x, the numerical coefficient is -2 and 15.

we can write the equation: -2x -15x + 4
then the numerical coefficients will be: -2 and -15
and 4 is the constant.

Question 2.
Underline the like terms.
Answer:
The like terms are defined as the terms that contain the same variable which is raised to the same power. In like terms, only the numerical coefficients can vary. We can combine the like terms to simplify the algebraic expressions so that the result of the expression can be obtained very easily.
Take the above-given expression:
-2x + 4 = 15 x
This equation can be written as: -2x -15x + 4
According to the definition, the like terms could be:
-2x – 15x
They are both like terms, so you can just simplify them.
-2x – 15x = -17x.

Question 3.
Which equation demonstrates the Distributive Property?
A. 3(n – 2) = 3n – 6
B. 8(n + 7) = 8(7 + n)
C. (n + 4) + 10 = n + 14
D. 12 + 3(n – 4) = 3(n – 4) + 12
Answer: Option A is correct.
Explanation:
What is Distributive Property?
The distributive property states that an expression which is given in form of A (B + C) can be solved as A × (B + C) = AB + AC. This distributive law is also applicable to subtraction and is expressed as, A (B – C) = AB – AC. This means operand A is distributed between the other two operands.
formula:
The distributive property formula of a given value is expressed as,
a(b + c) = ab + ac
According to the above definition and formula, option A is correct.
Therefore, 3(n – 2) = 3n – 6 is the correct.

Concepts and Skills

Question 4.
Use Tools The box shows how Andrew attempted to solve the equation \(\frac{1}{3}\)(n + 6) = -10. His work contains at least one error. List the step(s) with an error and find the correct solution. State what strategy and tool you will use to answer the question, explain your choice, and then find the answer.

Answer:
The above-given equation:1/3(n + 6) = -10
He used a distribution property
1/3(n) + 1/3(6) = -10
1/3(n) + 2 = -10
Now get ‘2’ to the right-hand side
1/3(n) = -10-2
1/3(n) = -12
if we get ‘1/3’ to the right-hand side then the equation will be:
n= -12/1/1/3
n=-12/1 x 3/1
n = -12 x 3
n = -36
This is the correct explanation.

Question 5.
Select whether each equation has no solution, one solution, or infinitely many solutions.

Answer:`
Now take the equation 1:
2.5(n + 4) = n + 1.5n – 7
there is no solution for this equation.
Now take the equation 2:
2.5(n + 4) = 2n + 0.5n + 10
Use the distributive property to multiply $2.5$ by $n+4$.
2.5n+10=2n+0.5n+10
combine 2n and 0.5n to get 2.5n
2.5n+10=2.5n+10
Subtract $2.5n$ from both sides.
2.5n+10−2.5n=10
Combine $2.5n$ and $-2.5n$ to get $0$.
10 = 10
compare 10 and 10
LHS = RHS
This is true for any n.
n ∈ R
This equation has infinite solutions.
Now take the equation 3:
2.5(n + 4) = 2.5n + 0.5n
Steps for solving linear equations:
Use the distributive property to multiply 2.5 by n+4.
2.5n+10=2.5n+0.5n
Combine 2.5n and 0.5n to get 3n.
2.5n+10=3n
Subtract ‘3n’ from both sides.
2.5n+10−3n=0
Combine 2.5n and −3n to get −0.5n.
−0.5n+10=0
Subtract ’10’ from both sides. Anything subtracted from zero gives its negation.
−0.5n=−10
n = -10/-0.5
Expand -10/-0.5 by multiplying both the numerator and the denominator by 10.
n = -100/-5
n = 20.
Therefore, In this equation, we get one solution.

Question 6.
Determine the value of b for which x = 1 is a solution of the equation shown.
2x+ 14= 10x + b
b = ____
Answer:
In the above-given equation, we need to find out the value of b.
The given equation:
2x+ 14= 10x + b
we know that x = 1 (given)
Now substitute the ‘x’ value in the given equation.
2(1) + 14 = 10(1) + b
2 + 14 = 10 + b
16 = 10 + b
Now get ’10’ to the left-hand side then the equation will be
16 – 10 = b
6 = b
Therefore, the value of b is 6.
Now substitute the ‘b’ value in the given equation.
2x + 14 = 10x + b
2(1) + 14 = 10(1) + 6
16 = 16
LHS = RHS
Hence, the answer is verified.

Solve each equation.

Question 7.
-3(x + 0.6) = 6(x + 2.4)
x = ____
Answer:
The given expression:
-3(x + 0.6) = 6(x + 2.4)
Steps for solving linear equation:
by using distributive property we solve the equation both sides.
-3x + (-1.8) = 6x + 14.4
-3x – 1.8 = 6x + 14.4
subtract 6x from both sides.
-3x – 1.8 – 6x = 6x + 14.4 – 6x
-3x – 1.8 – 6x = 14.4
-9x – 1.8 = 14.4
Now take ‘-1.8’ to the right-hand side
-9x = 14.4 + 1.8
-9x = 16.2
x = 16.2/-9
Expand 16.2/-9 by multiplying both numerator and denominator by 10
x = 162/-90
reduce the fraction 162/-90 to the lowest terms by extracting and cancelling out 18
x = -9/5
x = -1.8
Now substitute ‘x’ value in the above equation.
-3x – 1.8 = 6x + 14.4
-3(-1.8) – 1.8 = 6(-1.8) + 14.4
5.4 – 1.8 = -10.8 + 14.4
3.6 = 3.6
LHS = RHS
Hence, the answer is verified.

Question 8.
0.4(p – 5) = 0.6p + 2
p = ____
Answer:
The given equation:
0.4(p – 5) = 0.6p + 2
by using distributive property we simplify the equation.
0.4p – 0.4(5) = 0.6p + 2
0.4p – 2 = 0.6p + 2
Subtract $0.6p$ from both sides.
0.4p – 2 – 0.6p = 0.6p +2 – 0.6p
0.4p – 2 – 0.6p = 2
Combine $0.4p$ and $-0.6p$ to get $-0.2p$.
-0.2p – 2 = 2
Now get ‘-2’ to the right-hand side
-0.2p = 2 + 2
-0.2p = 4
p = 4/-0.2
expand 4/-0.2 by multiplying both numerator and the denominator by $10$.
p= 40/-2
p = -20
Now substitute ‘p’ value in the equation.
0.4(-20 – 5) = 0.6(-20) + 2
-10 = -12 + 2
-10 = -10
LHS = RHS
Hence, the answer is verified.

Question 9.
–\(\frac{1}{2}\)d + \(\frac{5}{8}\) = \(\frac{3}{8}\)d – \(\frac{11}{16}\)
d = ____
Answer:
The given expression:
1/2d + 5/8 = 3/8d – 11/16
get ‘3/8d’ to the left-hand side
1/2d+5/8-3/8d = -11/16
1/8d + 5/8 = -11/16
1/8d = -11/16 – 5/8
least common multiple of 16 and 8 is 16. Convert −11/16​ and 5/8​ to fractions with denominator $16$.
1/8d = -11/16 – 10/16
since -11/16 and 10/16 have the same denominator, subtract them by subtracting their numerators.
1/8d = -11-10/16
Subtract $10$ from $-11$ to get $-21$.
1/8d = -21/16
Multiply both sides by $8$, the reciprocal of 1/8.
d = -21/16 x 8
express -21/16 x 8 as a single fraction.
d = -21 x 8/16
d = -168/16
d = -21/2.

Question 10.
\(\frac{1}{4}\)(n + 7) = 5n – 7n + 1
n = ____
Answer:
The given equation:
1/4(n + 7) = 5n – 7n + 1
by using distributive property
1/4(n) + 1/4 x 7 = 5n – 7n + 1
1/4(n) + 7/4 = 5n – 7n + 1
combine 5n and -7n to get -2n
1/4(n) + 7/4 = -2n + 1
add 2n to the both sides
1/4(n) + 7/4 + 2n = 1
9/4(n) + 7/4 = 1
9/4(n) = 1-7/4
9/4(n) = 4-7/4
9/4(n) = -3/4
multiply both sides by 4/9, the reciprocal of 9/4
n = -3/4 x (4/9)
Multiply times 4/9 by multiplying numerator times numerator and denominator times denominator.
n = -3 x 4/4 x 9
n = -12/36
n = -1/3

Question 11.
Use numbers from the shaded box to complete the equation so that it has no solution.
3x + 2.5 = 12x – ___x + ____
Answer:

Question 12.
A rancher uses 280 feet of fencing to build a rectangular corral for a horse. The length of the corral is 2.5 times the width. What is the area of the corral?
A. 140 square feet
B. 700 square feet
C. 4,000 square feet
D. 16,000 square feet

Answer:
The fencing to build a rectangular corral for a horse is 280 feet
The length of the coral = 2.5 x w
for suppose if we take the width as 3
a = 2.5 x 280 = 700
area of rectangle = l x b
Therefore, the area of the corral is 700 square feet

Question 13.
Explain why the equation -2(3x + 6) = -6(x + 2) has infinitely many solutions.
Answer:
The given equation:
-2(3x + 6) = -6(x + 2)
-6x -12 = -6x – 12
-6x – 12 + 6x = -12
-12 = -12
LHS = RHS
This is true for any x
x∈R
x belongs to all real numbers. That’s why it has many infinite solutions.

Question 14.
Regular tickets cost $34.50, and VIP tickets cost $78.50. A total of 810 tickets were sold, and total ticket sales were $29,045. The equation 34.50r + 78.50(810 – r) = 29,045 can be used to determine the number of regular tickets r sold. How many of each type of ticket were sold?
Number of regular tickets: ____ Number of VIP tickets: _____
Answer:
The cost of a regular ticket = $34.50
The cost of a VIP ticket = $78.50
The number of tickets sold = 810
The total tickets sales = $29,045
The given equation:
34.50r + 78.50(810 – r) = 29,045
Use the distributive property to multiply $78.5$ by $810-r$.
34.5r + 63585 − 78.5r = 29045
Combine $34.5r$ and $-78.5r$ to get $-44r$.
−44r + 63585 = 29045
Subtract 63585 from both sides.
−44r = 29045 − 63585
−44r = −34540
r = -34540/-44
r = 785
The number of regular tickets = 785/34.50 = 23
The number of VIP tickets = 785/78.50 = 10

Question 15.
Two hikers walk on a trail in the direction of increasing mile marker numbers. Mandy starts at mile marker 1 and hikes at a rate of 2.5 miles per hour. At the same time, Rita starts at mile marker 2 and hikes at a rate of 3 miles per hour. The equation 2.5h + 1 = 3h + 2 represents the number of hours h it will take Mandy to catch up with Rita.
A. What is the solution of the equation?
h = _____
Answer:
The given equation:
2.5h + 1 = 3h + 2
2.5h + 1 − 3h = 2
− 0.5h + 1 = 2
−0.5h = 2 − 1
-0.5h = 1
h = 1/-0.5
Expand 1/-0.5​ by multiplying both the numerator and the denominator by 10.
h = 10/-5
h = -2

B. What does the solution of the equation indicate in this situation?
______________________
______________________
______________________
Answer:
the number of hours h it will take Mandy to catch up with Rita.
-2 hours it will take mandy to catch up with Rita
A negative h-value indicates a reversal in the directionality of the effect, which has no bearing on the significance of the difference between groups.