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Solve the equation justifying each step with the correct reasoning.

2(x + 8) = 2x + 8

Step 1: _______ Property to get Response area simplified equation

Step 2: _______ Property to get Response area simplified equation

lastly: For this equation, there is/are ____________

properties and reasons:
1. subtraction of equality
2. Multiplication of equality
3. combine like terms
4. division of equality
5. addition of equality
6. distributive

Equation simplified:
1. One solution
2. 16 = 8
3. 2x + 16 = 2x + 8
4. 2x + 8 = 2x + 8
5. No solution
6. x = -8
7. x + 16 = 8
8. 2x + 8 = 8
9. x = 2
10. x + 8 = 8

15 answers

Let's solve the equation step by step, justifying each with the correct reasoning.

Given equation:
\[ 2(x + 8) = 2x + 8 \]

Step 1: Use the Distributive Property to simplify the left side of the equation.
\[ 2(x + 8) = 2 \cdot x + 2 \cdot 8 \]
This simplifies to:
\[ 2x + 16 = 2x + 8 \]

Step 2: Subtract \( 2x \) from both sides using the Subtraction of Equality Property.
\[ 2x + 16 - 2x = 2x + 8 - 2x \]
This simplifies to:
\[ 16 = 8 \]

This statement, \( 16 = 8 \), is clearly false. Therefore, we conclude that the original equation has no solution.

Lastly, for this equation, there is/are no solution.

Summary of Properties and Reasoning:

  • Step 1: Distributive Property to simplify the equation to \( 2x + 16 = 2x + 8 \).
  • Step 2: Subtraction of Equality Property to derive \( 16 = 8 \).

Conclusion:

The equation simplifies to \( 16 = 8 \), which is not true. Therefore, there is no solution to the equation.

Match the description of the one variable equation with the number of solutions it will have.

1. x − 7 = 7 − x

2. 3(x + 5) = 3x + 5

3. 10 − x = 25

4. 2(x + 4) = 2x + 5 + 3

options:
1. one solution
2. Two solutions
3. No solution
4. Infinite solutions

Let's analyze each equation to determine how many solutions it has.

  1. Equation: \( x - 7 = 7 - x \)
    Solution:
    Adding \( x \) to both sides:
    \( x + x - 7 = 7 \)
    \[ 2x - 7 = 7 \]
    Adding 7 to both sides:
    \[ 2x = 14 \]
    Dividing by 2:
    \[ x = 7 \]
    Number of solutions: 1 (one solution)

  2. Equation: \( 3(x + 5) = 3x + 5 \)
    Solution:
    Distributing on the left:
    \( 3x + 15 = 3x + 5 \)
    Subtracting \( 3x \) from both sides:
    \( 15 = 5 \)
    This statement is false.
    Number of solutions: 3 (no solution)

  3. Equation: \( 10 - x = 25 \)
    Solution:
    Adding \( x \) to both sides:
    \( 10 = x + 25 \)
    Subtracting 25 from both sides:
    \( 10 - 25 = x \)
    \( -15 = x \)
    Thus, \( x = -15 \).
    Number of solutions: 1 (one solution)

  4. Equation: \( 2(x + 4) = 2x + 5 + 3 \)
    Solution:
    Distributing on the left:
    \( 2x + 8 = 2x + 8 \)
    This simplifies to:
    \( 2x + 8 - 2x = 8 \)
    Which gives us \( 8 = 8 \), a true statement.
    This indicates that the two sides are always equal for any value of \( x \).
    Number of solutions: 4 (infinite solutions)

Summary of Matches:

  1. \( x - 7 = 7 - x \) → 1 (one solution)
  2. \( 3(x + 5) = 3x + 5 \) → 3 (no solution)
  3. \( 10 - x = 25 \) → 1 (one solution)
  4. \( 2(x + 4) = 2x + 5 + 3 \) → 4 (infinite solutions)

So, the correct matches are:

  • 1 (one solution) for \( x - 7 = 7 - x \)
  • 3 (no solution) for \( 3(x + 5) = 3x + 5 \)
  • 1 (one solution) for \( 10 - x = 25 \)
  • 4 (infinite solutions) for \( 2(x + 4) = 2x + 5 + 3 \)
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?

A). 24n + 2 ≥ 37

B). 37 > 2n + 24

C). 37 ≥ 2n + 24

D). 37 < 2n + 24

To formulate the inequality that represents the student's situation, we need to account for the total cost of the calculator and the notebooks.

  1. The calculator costs $24.
  2. Each notebook costs $2, so if \( n \) represents the number of notebooks, then the cost for \( n \) notebooks would be \( 2n \).
  3. The total amount of money the student can spend is $37.

The total cost of the calculator and the notebooks together can be expressed as: \[ 24 + 2n \]

Since the student cannot spend more than $37, we can express this relationship as an inequality: \[ 24 + 2n \leq 37 \]

To re-arrange this, we can subtract 24 from both sides: \[ 2n \leq 37 - 24 \] \[ 2n \leq 13 \]

This means the inequality states that the total cost of the calculator and notebooks must be less than or equal to $37.

Now, looking at the answer choices provided:

A). \( 24n + 2 \geq 37 \)
B). \( 37 > 2n + 24 \)
C). \( 37 \geq 2n + 24 \)
D). \( 37 < 2n + 24 \)

The correct option that matches our derived inequality \( 24 + 2n \leq 37 \) can be rewritten as: \[ 37 \geq 2n + 24 \]

Thus, the correct answer is:
C). \( 37 ≥ 2n + 24 \)

Solve for b in the following equation: A = 1 over 2 (a + b)

A). b = 1 over 2A − a

B). b = 2A + a

C). b = 1 over 2A + a

D). b = 2A − a

To solve for \( b \) in the equation

\[ A = \frac{1}{2} (a + b), \]

we will follow these steps:

  1. Multiply both sides by 2 to eliminate the fraction: \[ 2A = a + b. \]

  2. Isolate \( b \) on one side of the equation by subtracting \( a \) from both sides: \[ b = 2A - a. \]

So, the solution is

\[ b = 2A - a. \]

Now, let's match this with the provided options:

  • A). \( b = \frac{1}{2}A - a \)
  • B). \( b = 2A + a \)
  • C). \( b = \frac{1}{2}A + a \)
  • D). \( b = 2A - a \)

The correct answer is:
D). \( b = 2A - a \).

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?

A). Substitution Property

B). Combine Like Terms

C). Addition Property of Equality

D). Distributive Property

E). Commutative Property of Addition

To analyze the steps provided and identify the property used to go from step 3 to step 4, we can take a closer look at those specific steps:

  • Step 3: \( 5x - 6 = 19 \)
  • Step 4: \( 5x - 6 + 6 = 19 + 6 \)

In step 4, the student adds 6 to both sides of the equation to isolate \( 5x \).

Thus, the property used here is:

C). Addition Property of Equality.

This is because the Addition Property of Equality states that if you add the same value to both sides of an equation, the equality remains valid.

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

1. Identify the property used to arrive at Step 1: _____________

2. What step includes a mistake made from the previous step? ___________

3. What should have been the correct answer for x ? ____________

Let’s go through the provided steps and address each part of your question:

Original equation: \[ x + 2(x + 1) = 17 \]

Step 1:
\[ x + 2x + 2 = 17 \]
This step combines the terms from the left-hand side.

  1. Identify the property used to arrive at Step 1:
    The property used in Step 1 is the Distributive Property. When the student multiplies 2 by both \( x \) and \( 1 \) in the expression \( 2(x + 1) \), they distribute the 2 to get \( 2x + 2 \).

Step 2:
\[ 3x + 2 = 17 \]
This step is correct, as \( x + 2x \) correctly combines to form \( 3x \).

Step 3:
\[ 3x = 15 \]
This is also correct, as it results from subtracting 2 from both sides of the equation:

From \( 3x + 2 = 17 \), subtracting 2 gives: \[ 3x + 2 - 2 = 17 - 2 \] which simplifies to: \[ 3x = 15 \]

Step 4:
\[ x = 45 \]
This step includes a mistake. To solve for \( x \), the last step should have involved dividing both sides by 3.

  1. What step includes a mistake made from the previous step?
    Step 4 includes the mistake. The student incorrectly calculated \( x \) by not dividing \( 15 \) by \( 3 \). The correct operation should be: \[ x = \frac{15}{3} \]

What should have been the correct answer for \( x \)?
The correct calculation for \( x \) is: \[ x = \frac{15}{3} = 5 \]

So:

  • 1. Identify the property used to arrive at Step 1: Distributive Property
  • 2. What step includes a mistake made from the previous step? Step 4
  • 3. What should have been the correct answer for \( x \)? 5
what's the last answer again?

The correct answer for \( x \) should be:

5.

This is obtained by solving \( 3x = 15 \) as follows: \[ x = \frac{15}{3} = 5. \]

so x = 5

Yes, that's correct! The solution to the equation \( x + 2(x + 1) = 17 \) is \( x = 5 \).

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