16. 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 ax - 5 = 18
We need to isolate the variable x, so we can start by adding 5 to both sides of the equation:
ax - 5 + 5 = 18 + 5
ax = 23
To solve for x, we need to divide both sides of the equation by a:
(ax)/a = 23/a
x = 23/a
So the solution to the equation is x = 23/a.
Part B: Solve the inequality ax - 5 > 18
We need to isolate the variable x, so we can start by adding 5 to both sides of the inequality:
ax - 5 + 5 > 18 + 5
ax > 23
To solve for x, we need to divide both sides of the inequality by a. However, we need to be cautious when dividing by a negative number. If a is negative, we need to flip the inequality sign. If a is positive, we keep the inequality sign as it is.
Case 1: a > 0
(ax)/a > 23/a
x > 23/a
Case 2: a < 0
(ax)/a < 23/a
x < 23/a
But since we flipped the inequality sign, we need to reverse the direction of the inequality:
x > 23/a
So the solution to the inequality depends on the value of a. If a is positive, the solution is x > 23/a. If a is negative, the solution is x < 23/a.
Part C: The difference between an equation and an inequality regarding solving steps and solutions is as follows:
Solving Steps:
- In an equation, the goal is to find the value(s) of the variable that make the equation true.
- In an inequality, the goal is to find the range of values for the variable that satisfy the inequality.
Solutions:
- In an equation, there can be one or multiple solutions, or even no solution.
- In an inequality, there can be a range of values that satisfy the inequality, represented by an interval or inequality notation.
1. Which equation has the same solution for x as this equation: x−12=40
?(1 point)
Responses
x2=14
x over 2 is equal to 14
x+12=−40
x plus 12 is equal to negative 40
x4=13
x over 4 is equal to 13
12−x=40
12 minus x is equal to 40
Question 2
2. 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)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
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
Question 3
3. Match the equation with its solution(s).(5 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
3(2i−7)=15
3i+2i−7=18
3i+5=2i−7
3i+5=3i+7
3(2i+7)=6i+21
Question 4
4. Solve the equation justifying each step with the correct reasoning.
2(x+8)=2x+8
(5 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
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
Properties and Reasons
Equation simplified
Question 5
5. Match the description of the one variable equation with the number of solutions it will have.(4 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
x−7=7−x
3(x+5)=3x+5
10−x=25
2(x+4)=2x+5+3
Question 6
6. 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
24n+2≥37
24 n plus 2 is greater than or equal to 37
37>2n+24
37 is greater than 2 n plus 24
37<2n+24
37<2n+24
37≥2n+24
37 is greater than or equal to 2 n plus 24
Question 7
7. Solve for b in the following equation: A=12(a+b)
(1 point)
Responses
b=2A−a
b is equal to 2 cap A minus A
b=2A+a
b is equal to 2 cap A plus A
b=12A−a
b is equal to 1 half cap A minus A
b=12A+a
b is equal to 1 half cap A plus A
Question 8
8. Graph the solutions for the inequality: −3x+1≤−47
(2 points)
Responses
Question 9
9. 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 should have filled in the point at -4 to show the solution x could be equal to -4
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 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
The student did not make a mistake; this is the correct graph of the inequality
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 x is less than -4, when the variable is on the other side; -4 is less than x so x is greater than -4
Question 10
10. 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
Combine Like Terms
Combine Like Terms
Substitution Property
Substitution Property
Commutative Property of Addition
Commutative Property of Addition
Addition Property of Equality
Addition Property of Equality
Distributive Property
Distributive Property
Question 11
11. 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
(3 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
Identify the property used to arrive at Step 1: Response area
What step includes a mistake made from the previous step? Response area
What should have been the correct answer for x ? Response area
Question 12
12. 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:
Question 13
13. 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 multiply by 9 on each side.
First step is to add 32 to both sides.
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 multiply both sides by 5
The next step from F - 32 = 9/5 C, could be to divide 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
Solution: 5/9F - 32 = C
To solve this equation for C, you would have to distribute at some point.
To solve this equation for C, you would have to distribute at some point.
Question 14
14. Izzie 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 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 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
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 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
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
Question 15
15. Which graph represents the solutions to this inequality: 5n−3<3(n+1)
(2 points)
Responses
Question 16
16. 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.
(6 points)
Part A: Solve the equation ax - 5 = 18
To isolate the variable x, we can start by adding 5 to both sides of the equation:
ax - 5 + 5 = 18 + 5
ax = 23
Next, we divide both sides of the equation by a:
(ax)/a = 23/a
x = 23/a
So the solution to the equation ax - 5 = 18 is x = 23/a.
Part B: Solve the inequality ax - 5 > 18
To solve the inequality, we need to isolate the variable x. We can start by adding 5 to both sides of the inequality:
ax - 5 + 5 > 18 + 5
ax > 23
Next, we divide both sides of the inequality by a. However, if a is negative, we need to flip the inequality sign. So we have two cases:
Case 1: a > 0
(ax)/a > 23/a
x > 23/a
Case 2: a < 0
(ax)/a < 23/a
x < 23/a
So the solution to the inequality ax - 5 > 18 depends on the value of a. If a is positive, the solution is x > 23/a. If a is negative, the solution is x < 23/a.
Part C: The difference between an equation and an inequality regarding solving steps and solutions is as follows:
Solving Steps:
- In an equation, the goal is to find the value(s) of the variable that make the equation true.
- In an inequality, the goal is to find the range of values for the variable that satisfy the inequality.
Solutions:
- In an equation, there can be one or multiple solutions, or even no solution.
- In an inequality, there can be a range of values that satisfy the inequality, represented by an interval or inequality notation.
Part A: Solve the equation ax - 5 = 18.
Step 1: Start by isolating the variable term by adding 5 to both sides of the equation:
ax - 5 + 5 = 18 + 5
ax = 23
Step 2: To solve for x, divide both sides of the equation by the coefficient of x (a):
(ax) / a = 23 / a
x = 23 / a
The solution to the equation ax - 5 = 18 is x = 23 / a.
Part B: Solve the inequality ax - 5 > 18.
Step 1: Again, start by isolating the variable term by adding 5 to both sides of the inequality:
ax - 5 + 5 > 18 + 5
ax > 23
Step 2: To solve for x, divide both sides of the inequality by the coefficient of x (a). It is important to note that if a is negative, we need to reverse the direction of the inequality:
(ax) / a > 23 / a
If a is positive: x > 23 / a
If a is negative: x < 23 / a
The solution to the inequality ax - 5 > 18 is x > 23 / a if a is positive, or x < 23 / a if a is negative.
Part C: The difference between an equation and an inequality lies in the symbols used and the nature of the solutions.
An equation uses an equal sign (=) and represents a statement of equality between two expressions, while an inequality uses symbols such as >, <, ≥, or ≤ and represents a statement of the relationships between two expressions regarding their sizes or values.
When solving an equation, the goal is to find a specific value (or values) of the variable that makes the equation true.
When solving an inequality, the goal is to find a range of values for the variable that satisfies the given condition. The solution to an inequality is typically expressed as a set of possible values or an interval, rather than a single value.
The solving steps for equations and inequalities can be similar in some cases, but inequalities may involve additional steps, such as reversing the direction of the inequality when multiplying or dividing by a negative number.
In summary, equations are used to find specific values of a variable that make the equation true, while inequalities are used to find a range of values that satisfy the given condition.
Part A: Solve the equation for x showing all steps for credit: ax-5=18
To solve the equation, we need to isolate the variable x on one side of the equation. We can do this by performing the same operation on both sides of the equation.
Step 1: Add 5 to both sides of the equation:
ax - 5 + 5 = 18 + 5
This simplifies to:
ax = 23
Step 2: Divide both sides of the equation by a:
ax / a = 23 / a
This simplifies to:
x = 23/a
Therefore, 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
To solve the inequality, we need to isolate the variable x on one side of the inequality sign. Again, we perform the same operation on both sides of the inequality.
Step 1: Add 5 to both sides of the inequality:
ax - 5 + 5 > 18 + 5
This simplifies to:
ax > 23
Step 2: Divide both sides of the inequality by a. However, since we don't know if a is positive or negative, we need to consider two cases:
Case 1: a > 0
If a is positive, then dividing by a does not change the direction of the inequality:
x > 23/a
Case 2: a < 0
If a is negative, then dividing by a reverses the direction of the inequality:
x < 23/a
Therefore, the solutions for the inequality ax-5 > 18 are x > 23/a (if a > 0) and x < 23/a (if a < 0).
Part C: State the difference between an equation and an inequality regarding solving steps and solutions.
An equation is a mathematical statement that shows that two expressions are equal. When solving an equation, our goal is to find the specific value(s) of the variable that make both sides of the equation equal. To solve an equation, we perform operations on both sides of the equation while maintaining the equality.
On the other hand, an inequality is a mathematical statement that indicates a relationship between two expressions, typically denoted by symbols such as < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). While solving an inequality, our goal is to find the range of values for the variable that satisfy the given inequality. To solve an inequality, we perform operations on both sides of the inequality, but the direction of the inequality sign might change depending on the operations performed and the sign of the coefficients.
In summary, the main difference between an equation and an inequality lies in the final solutions. An equation has one or more specific values that make the equation true, while an inequality may have a range of values that satisfy the given inequality.