Equivalent Expressions
Explain in words why 8(2x + 1) is equivalent to 16x + 8.
Distributive property: multiply 8 by both 2x and 1
What is the perimeter of a square with side length x?
x+x+x+x = 4x
Solve: 2x − 3 = 7
x = 5
Solve: 4x−7=x+11
x = 6
18g−12≤75
g≤4.83
Which expression(s) are equivalent to 12y + 8? Explain.
a. 12(y + 8)
b. 6(2y + 2) - 4
c. 10(2y - 2) + 8y
d. 4(3y + 2)
c. 6(2y + 2) - 4
d. 4(3y + 2)
What is the area of a rectangle with length of 4 and width x + 3?
4(x+3) = 4x + 12
Solve: -2x + 5 = -11
x = 8
4.5x+2=2.5x+12
x = 5
Kayla burns 8 calories per minute while practicing basketball and starts with a warm-up that burns 50 calories. She wants to burn at least 250 calories by the end of practice.
Write and solve an inequality to represent how many minutes, m, she should practice.
8m + 50 ≥ 250
m≥25
Simplify the expression:
4x - 16 + 2 - 6x
-2x - 14
A rectangle is 5 units longer than it's width, x. Write an expression for its perimeter.
4x + 10
2x + 2(x+5) = 2x + 2x + 10 = 4x + 10
or x + x + (x + 5) + (x +5) = 4x + 10
A taxi charges $3 plus $2 per mile. If the ride cost $19, how many miles did you travel? Create and solve an equation.
8 miles
3 + 2m = 19
Solve: 5(x - 2) - 3 = 17
x = 6
A group is planning a movie night. The cost to reserve the theater is $30, plus $4 per person attending. The group budget is no more than $110.
Write and solve an inequality to show how many people, p, can attend without going over the budget.
30 + 4p ≤ 100
p≤20
Simplify the expression:
6x - 3(x-1)
3x + 3
The area of a rectangular field is (30𝑥+1,800) square feet. Which expression BEST shows the dimensions of the field?
a. 30(𝑥+1,800)
b. 3(10𝑥+600)
c. 45(𝑥+40)
d. 30(𝑥+60)
d. 30(x+60)
Explanation: Even though b also expands to the same area, it gives a dimension of 3 feet, which would mean the field is only 3 feet wide — not reasonable for a field or garden.
But d gives 30 feet by (x + 60) feet, which makes sense as a pair of actual garden dimensions.
A streaming service charges $5 per month plus a one-time activation fee of $8. If you paid $33 total, how many months did you subscribe? Create and solve the equation.
5 months
5m + 8 = 33
A babysitter charges $8 per hour plus a $6 flat fee. Another babysitter charges $5 per hour and charges $10 for flat fee. How many hours did the babysitters work for their cost to be the same? (Write your answer as a mixed number)
1 1/3 hours
8h + 6 = 5h + 10
h= 4/3 or 1.333...
Nick earns $6 for each dog he walks and a $10 bonus from his neighbor. He wants to earn at least $40 this weekend. Write and solve an inequality to find how many dogs he must walk.
6d + 10 ≥ 40
d ≥ 5
Simplify the expression:
-8(x - 3x)
-8x + 24x = 16x
The expression (2x + 16) + (2x + 8) represents the perimeter of a rectangular walkway. Which expression shows the perimeter of the walkway in terms of the side lengths?
a. 2(x+8) + 2(x+4)
b. 4(x+6)
c. 4(x + 24)
d. 2(x+4) + 2(x+2)
a. 2(x+8) + 2(x+4)
Explanation: A rectangle has two of the same lengths and two of the same widths. So this expression shows that (x+8) and (x+4) are the length and width; also they correctly simplify to the original expression.
b. 4(x+6) and c. 4(x+24) represent squares that have 4 of the same side lengths.
Solve for m:
pm + x = 100
m = (100 - x)/p
A landscaper charges $12.50 per hour plus a $15 supply fee. Another landscaper charges $10.75 per hour plus a $25 supply fee. After x hours, the cost is the same. Solve for x. (Round to the nearest tenths)
5.7 hours
12.50h + 15 = 10.75h + 25
X = 5.7
Plan A costs $30 for service fee plus $6 per fitness class.
Plan B costs $50 for service fee plus $3 per fitness class.
Maya wants to know for what number of classes Plan A is the better deal (costs less) than Plan B.
Write and solve an inequality. (Write the number of months as a whole number)
30+6c < 50 + 3c
c < 6.666…
Less than 6 months