Which equation has the same solution for x as this equation: x−12=40
?(1 point)
Responses
x2=14
x4=13
x+12=−40
12−x=40
x+12=-40
. The length of a rectangle is four meters less than twice its width.
If the area of the rectangle is 96 m^2, what is the length and the width?
(3 points)
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An expression for the length of the rectangle in terms of the width would be Response area
The formula for the area of a rectangle is Response area
Using trial and error, if the area is 96 m^2, then the length and width are Response area
An expression for the length of the rectangle in terms of the width would be 2w - 4.
The formula for the area of a rectangle is length * width.
Using trial and error, if the area is 96 m^2, then the length and width are 12 m and 8 m respectively.
Match the equation with its solution(s).(5 points)
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3(2i−7)=15
3i+2i−7=18
3i+5=3i+7
3(2i+7)=6i+21
3i+5=2i−7
3(2i−7)=15 - Solution: i = 6
3i+2i−7=18 - Solution: i = 5
3i+5=3i+7 - No solution
3(2i+7)=6i+21 - Solution: i = -1
3i+5=2i−7 - Solution: i = -12
Solve the equation justifying each step with the correct reasoning.
2(x+8)=2x+8
(5 points)
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Step 1: Response area Property to get Response area simplified equation
Step 2: Response area Property to get Response area simplified equation
For this equation, there is/are Response area
Step 1: Distribute the 2 to both terms inside the parentheses.
2(x+8) = 2x + 2(8)
Step 2: Simplify the right side of the equation.
2(x+8) = 2x + 16
For this equation, there is no further simplification or solving required. The simplification in Step 2 is the final solution.
Match the description of the one variable equation with the number of solutions it will have.(4 points)
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x−7=7−x
3(x+5)=3x+5
10−x=25
2(x+4)=2x+5+3
x−7=7−x - One solution
3(x+5)=3x+5 - Infinite solutions
10−x=25 - No solution
2(x+4)=2x+5+3 - No solution
A student wants to purchase some new school supplies. He wants to buy a calculator that costs $24 and some notebooks for school. Each notebook costs $2. The student only has $37 to spend.
Let n represent the number of notebooks that he buys.
Which inequality describes this scenario?
(1 point)
Responses
37≥2n+24
37<2n+24
37>2n+24
24n+2≥37
37≥2n+24
Solve for b in the following equation: A=12(a+b)
(1 point)
Responses
b=2A+a
b=12A+a
b=12A−a
b=2A−a
b= (A - 12a)/12
Solve for b in the following equation: A=12(a+b)
(1 point)
Responses
A) b=2A+a
B) b=12A+a
C) b=12A−a
D) b=2A−a
A) b=2A+a
A student claims that graph below represents the solutions to the inequality: −4<x
What was the student's mistake?
(1 point)
Responses
The student should have filled in the point at -4 to show the solution x could be equal to -4
The student did x is less than -4, when the variable is on the other side; -4 is less than x so x is greater than -4
The student did not make a mistake; this is the correct graph of the inequality
The student should have multiplied by a negative and switched the direction of the arrow on the graph to go right instead of left
The student did not make a mistake; this is the correct graph of the inequality.
A student solves the following equation:
Problem: 2(x−3)+3x=19
Step 1: 2x−6+3x=19
Step 2: (2x+3x)−6=19
Step 3: 5x−6=19
Step 4: 5x−6+6=19+6
Step 5: 5x=25
Step 6: x=5
What property justifies going from step 3 to step 4?
(1 point)
Responses
Commutative Property of Addition
Addition Property of Equality
Distributive Property
Combine Like Terms
Substitution Property
Step 4: 5x−6+6=19+6
The property that justifies going from step 3 to step 4 is the Addition Property of Equality.
A student solved the equation: x+2(x+1)=17
Step 1: x+2x+2=17
Step 2: 3x+2=17
Step 3: 3x=15
Step 4: x=45
The solution is incorrect. Let's go through the steps and identify the mistake:
Step 1: x+2x+2=17 - Correct
Step 2: 3x+2=17 - Correct
Step 3: 3x=15 - Correct
Step 4: x=45 - Mistake
The mistake is in Step 4. To isolate x and solve for it, we need to divide both sides of the equation by 3, not 45. The correct solution is:
Step 4: x = 15/3 = 5
So, the correct solution to the equation is x = 5.
Identify the property used to arrive at Step 1:
The property used to arrive at Step 1 is the Distributive Property.
What should have been the correct answer for x ? Response area
The correct answer for x is 5.
Grandma is removing weeds from her garden. She estimates that there are 250 weeds in the garden at the moment, and she can remove weeds at a rate of 5 per minute. At this rate, how many minutes will it take her to only have 30 weeds remaining in the garden?(3 points)
Equation:
Solution:
Meaning of the solution in words:
Equation: 250 - 5x = 30
Solution: x = 44
Meaning of the solution in words: It will take Grandma 44 minutes to remove enough weeds from her garden to have only 30 weeds remaining.
The Celsius to Fahrenheit formula: F=95C+32
Solve this formula for C. Check all answers that are true.
(4 points)
Responses
First step is to multiply by 9 on each side.
First step is to add 32 to both sides.
The next step from F - 32 = 9/5 C, could be to multiply both sides by 5
The next step from F - 32 = 9/5 C, could be to divide both sides by 5
Solution: 5/9F - 32 = C
To solve this equation for C, you would have to distribute at some point.
First step is to subtract 32 from both sides.
The next step from F - 32 = 9/5C could be to multiply both sides by 5/9.
Solution: C = 5/9(F - 32)
To solve this equation for C, you would have to distribute at some point.
zzie is trying to solve this inequality: 3(t+1)−4t≥−5
After she solves it, she shows it to you and asks you if she did it correctly.
This is the work she completed:
Step 1: 3t+3−4t≥−5
Step 2: −t+3≥−5
Step 3: −t≥−2
Step 4: t≤2
(12 points)
True False
Izzie made a mistake going from step 2 to step 3
True – Izzie made a mistake going from step 2 to step 3
False – Izzie made a mistake going from step 2 to step 3
The correct answer should have been t≤8
True – The correct answer should have been t is less than or equal to 8
False – The correct answer should have been t is less than or equal to 8
Izzie should not have changed the inequality symbol from ≥
to ≤
True – Izzie should not have changed the inequality symbol from is greater than or equal to to is less than or equal to
False – Izzie should not have changed the inequality symbol from is greater than or equal to to is less than or equal to
Izzie made a mistake going from step 3 to step 4
True – Izzie made a mistake going from step 3 to step 4
False – Izzie made a mistake going from step 3 to step 4
Izzie made a mistake going from step 1 to step 2
True – Izzie made a mistake going from step 1 to step 2
False – Izzie made a mistake going from step 1 to step 2
Izzie made no mistakes in solving this inequality
True – Izzie made no mistakes in solving this inequality
False – Izzie made no mistakes in solving this inequality
False – Izzie made a mistake going from step 2 to step 3
The correct answer should have been t ≤ 8: False – The correct answer should have been t≤2
Izzie should not have changed the inequality symbol from ≥ to ≤: False – Izzie should not have changed the inequality symbol from ≥ to ≤
Izzie made a mistake going from step 3 to step 4: False – Izzie made a mistake going from step 3 to step 4
Izzie made a mistake going from step 1 to step 2: True – Izzie made a mistake going from step 1 to step 2
Izzie made no mistakes in solving this inequality: False – Izzie made no mistakes in solving this inequality
Solve the equation and inequality for x.
Part A: Solve the equation for x showing all steps for credit: ax−5=18
Part B: Solve the inequality for x showing all steps for credit: ax−5>18
Part C: State the difference between an equation and an inequality regarding solving steps and solutions.
Part A: Solve the equation for x showing all steps for credit: ax−5=18
Step 1: Add 5 to both sides of the equation.
ax−5+5=18+5
ax=23
Step 2: Divide both sides of the equation by a.
(ax)/a = 23/a
x = 23/a
So, the solution for the equation ax−5=18 is x = 23/a.
Part B: Solve the inequality for x showing all steps for credit: ax−5>18
Step 1: Add 5 to both sides of the inequality.
ax−5+5>18+5
ax>23
Step 2: Divide both sides of the inequality by a.
(ax)/a > 23/a
x > 23/a
So, the solution for the inequality ax−5>18 is x > 23/a.
Part C: The difference between an equation and an inequality regarding solving steps and solutions is as follows:
1. Equations: When solving an equation, the goal is to find a value (or values) of the variable that makes the equation true. The steps involve simplifying both sides of the equation and isolating the variable. The solution to an equation is typically a specific value or set of values that satisfy the equation.
2. Inequalities: When solving an inequality, the goal is to find a range of values that make the inequality true. The steps involve simplifying both sides of the inequality and isolating the variable, just like in equations. However, when solving an inequality, there are often additional steps, such as flipping the inequality sign when multiplying or dividing by a negative number. The solution to an inequality is a range of values that satisfy the inequality and can be represented using inequality notation, such as x > 5 or x < 10.