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Question 1 A) Use the following step as a guide to solve the following equation. Write out your work for all steps to solve the equation. 2x - 5 = 3(4x + 5) (8 points) BoldItalicUnderlineBullet listNumbered listΣ 0 / 10000 Word Limit Question 2 A)Solve the equation 12=bY−312=bY−3 for Y. Show each step used to solve for Y and justify by giving the property used for each step. (6 points) BoldItalicUnderlineBullet listNumbered listIncrease indentDecrease indentΣ 0 / 10000 Word Limit Question 3 A) Gwen volunteered to work at the ticket booth for her school's Halloween carnival. The chart below gives the number of hours Gwen worked and the total number of tickets she sold. Hours (h) Tickets (t) 1 23 2 46 3 69 4 92 Based on the table, write an equation for the relation between the number hours Gwen worked and the number of tickets she sold. (1 point) Responses ht=23ht=23h t is equal to 23 h=23th=23th is equal to 23 t t=h23t=h23t is equal to h over 23 t=23ht=23ht is equal to 23 h Question 4 A)(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. Use the equation y−6=x4y−6=x4 to fill in the missing values in the table below. x y 32 14 12 Response area 0 Response area Response area 4 Response area -3 -81269-9-3610 Question 5 A) Solve: −8+6x−(−12)=6x−16+2x−8+6x−(−12)=6x−16+2x(1 point) Responses No Real Solutions No Real Solutions x=−18x=−18x is equal to negative 18 x=−6x=−6x is equal to negative 6 x=10x=10x is equal to 10 Question 6 A) Identify the steps followed to solve the equation 5−3(x+3)=11−8x5−3(x+3)=11−8x(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. 5−3x−9=11−8x5−3x−9=11−8x −3x−4=11−8x−3x−4=11−8x 5x−4=115x−4=11 5x=155x=15 x=3x=3 Distributive PropertyCombine Like TermsMultiplication Property of EqualitiesDivision Property of EqualitiesAddition Property of EqualitiesSubtraction Property of Equalities Question 7 A) A student solved the equation x+2(x+1)=17x+2(x+1)=17 below and showed their work in the table below. Problem x+2(x+1)=17x+2(x+1)=17 Line 1 x+2x+2=17x+2x+2=17 Line 2 3x+2=173x+2=17 Line 3 3x=153x=15 Line 4 x=5x=5 Identify the property used to arrive at Line 1. (1 point) Responses Subtraction Property of Equalities Subtraction Property of Equalities Collect Like Terms Collect Like Terms Distributive Property Distributive Property Addition Property of Equalities Addition Property of Equalities Question 8 A) Solve. y7+8=22y7+8=22(1 point) Responses y=98y=98y is equal to 98 y=2y=2y is equal to 2 y=7y=7y is equal to 7 y=30y=30y is equal to 30 Question 9 A) Solve. 18x−3=218x−3=2 Which of the following equations has the same solution as the equation above? (1 point) Responses 2x−3=182x−3=182 x minus 3 is equal to 18 x−4=14x−4=14x minus 4 is equal to 14 2x−6=182x−6=182 x minus 6 is equal to 18 x−318=2x−318=2the fraction with numerator x minus 3 and denominator 18 is equal to 2 Question 10 A) The sum of three consecutive even integers (3 even integers in a row) is 120. Use the following setup and given equation to find your solution: 1st integer = xx 2nd integer = x+2x+2 3rd integer = x+4x+4 Equation: x+(x+2)+(x+4)=120x+(x+2)+(x+4)=120 Solve the equation above. What is the largest of the three integers? (1 point) Responses 38 38 44 44 42 42 38, 42, 44 38, 42, 44 Question 11 A) The area of a triangle can be found using the formula: A=12hbA=12hb Solve the area formula for b. (1 point) Responses 12b=Ah12b=Ah1 half b is equal to cap A over h b=A2hb=A2hb is equal to cap A over 2 h b=2Ahb=2Ahb is equal to 2 cap A over h b=2(A−h)b=2(A−h)b is equal to 2 times open paren cap A minus h close paren Question 12 A) Solve. 7(5x+2)=−4(6−5x)7(5x+2)=−4(6−5x)(1 point) Responses x=1538x=1538x is equal to 15 over 38 x=3815x=3815x is equal to 38 over 15 x=−3815x=−3815x is equal to negative 38 over 15 x = −1538x = −1538x = −1538x = −1538 Question 13 A) Two buses start at the same point and travel in opposite directions. The first bus travels 10 miles per hour faster than the second bus. In 3 hours, the buses are 282 miles apart. Use the formula below to determine the rate of the second bus. 3(r + 10) + 3r = 282 (1 point) Responses r = 45 mph r = 45 mph r = 42 mph r = 42 mph r = 30 mph r = 30 mph r = 21 mph r = 21 mph Question 14 A) The formula for the area of a triangle is A=bh2A=bh2 Which equation show the formula in terms of b? (1 point) Responses h=2Abh=2Abh is equal to 2 cap A over b b=2Ahb=2Ahb is equal to 2 cap A over h h=b2Ah=b2Ah is equal to b over 2 cap A b=h2Ab=h2Ab is equal to h over 2 cap A Question 15 A)What is the solution set of 2x+3≥7 and 2x≥82x+3≥7 and 2x≥8(1 point) Responses {x|x≥2}{x|x≥2}start set x such that x is greater than or equal to 2 end set {x|x≥4}{x|x≥4}start set x such that x is greater than or equal to 4 end set {x|x≤2}{x|x≤2}start set x such that x is less than or equal to 2 end set {x|x≤4}{x|x≤4}start set x such that x is less than or equal to 4 end set Question 16 A)What is the solution set of 3x−2=73x−2=7 and 4x+5≤21and 4x+5≤21(1 point) Responses {x|x=3}{x|x=3}start set x such that x is equal to 3 end set {x|x≤3}{x|x≤3}start set x such that x is less than or equal to 3 end set {x|x=4}{x|x=4}start set x such that x is equal to 4 end set {x|x≤4}{x|x≤4}start set x such that x is less than or equal to 4 end set

11 answers

Question 1 A)

To solve the equation \(2x - 5 = 3(4x + 5)\), follow these steps:

  1. Distribute on the right side: \[ 2x - 5 = 3 \cdot 4x + 3 \cdot 5 \] \[ 2x - 5 = 12x + 15 \]

  2. Move all terms involving \(x\) to one side and constant terms to the other side. First, let's subtract \(12x\) from both sides: \[ 2x - 12x - 5 = 15 \] This simplifies to: \[ -10x - 5 = 15 \]

  3. Now, add \(5\) to both sides: \[ -10x - 5 + 5 = 15 + 5 \] \[ -10x = 20 \]

  4. Finally, divide by \(-10\): \[ x = \frac{20}{-10} \] \[ x = -2 \]

Final answer: \(x = -2\)

Question 2 A)

To solve the equation \(12 = bY - 3\) for \(Y\), follow these steps:

  1. Add \(3\) to both sides (to isolate the term containing \(Y\)): \[ 12 + 3 = bY \] \[ 15 = bY \] (Addition Property of Equality)

  2. Now divide both sides by \(b\) (to solve for \(Y\)): \[ Y = \frac{15}{b} \] (Division Property of Equality)

Final answer: \(Y = \frac{15}{b}\)

Question 3 A)

To derive the equation for the relationship between the number of hours \(h\) Gwen worked and the number of tickets \(t\) she sold, we notice the pattern in the values provided:

  • For every hour \(h\), Gwen sold \(23\) tickets.
  • Therefore, the total tickets sold \(t\) can be represented as a function of hours worked \(h\):

\[ t = 23h \]

Final answer: \(t = 23h\)

Question 4 A)

To fill in the missing values in the table using the equation \(y - 6 = x \cdot 4\), we can rearrange it to:

\[ y = 4x + 6 \]

Now we can plug in the values for \(x\):

  1. For \(x = 3\): \[ y = 4(3) + 6 = 12 + 6 = 18 \]

  2. For \(x = 2\): \[ y = 4(2) + 6 = 8 + 6 = 14 \]

  3. For \(x = 4\): \[ y = 4(4) + 6 = 16 + 6 = 22 \]

  4. For \(x = -3\): \[ y = 4(-3) + 6 = -12 + 6 = -6 \]

Final table filled: \[ \begin{array}{|c|c|} \hline x & y \ \hline 3 & 18 \ 2 & 14 \ 4 & 22 \ -3 & -6 \ \hline \end{array} \]

Question 5 A)

To solve the equation \(-8 + 6x - (-12) = 6x - 16 + 2x\):

  1. Rewrite the equation: \[ -8 + 6x + 12 = 6x - 16 + 2x \]

  2. Combine like terms: \[ 4 + 6x = 8x - 16 \]

  3. Move \(6x\) to the right: \[ 4 = 8x - 6x - 16 \] \[ 4 = 2x - 16 \]

  4. Add \(16\) to both sides: \[ 4 + 16 = 2x \] \[ 20 = 2x \]

  5. Divide both sides by \(2\): \[ x = 10 \]

Final answer: \(x = 10\)

Question 6 A)

Identify the steps followed to solve the equation \(5 - 3(x + 3) = 11 - 8x\):

  1. Distributive Property: \[ 5 - 3x - 9 = 11 - 8x \]

  2. Combine Like Terms: \[ -3x - 4 = 11 - 8x \]

  3. Addition Property of Equality (adding \(8x\)): \[ 5x - 4 = 11 \]

  4. Addition Property of Equality (adding \(4\)): \[ 5x = 15 \]

  5. Division Property of Equality: \[ x = 3 \]

Question 7 A)

Identify the property used to arrive at Line 1:

The equation is \(x + 2(x + 1) = 17\).

In Line 1, we distribute \(2\) through \( (x + 1) \): \[ x + 2x + 2 = 17 \] Property used: Distributive Property

Question 8 A)

Solve the equation: \[ \frac{y}{7} + 8 = 22 \]

  1. Subtract \(8\) from both sides: \[ \frac{y}{7} = 14 \]

  2. Multiply both sides by \(7\): \[ y = 98 \]

Final answer: \(y = 98\)

Question 9 A)

Solve \(18x - 3 = 2\):

  1. Add \(3\): \[ 18x = 5 \]

  2. Divide by \(18\): \[ x = \frac{5}{18} \]

Which equation has the same solution?

Final Answer: \(x - 3 = 18\)

Question 10 A)

The setup is: \[ x + (x + 2) + (x + 4) = 120 \]

  1. Combine: \[ 3x + 6 = 120 \]

  2. Subtract \(6\): \[ 3x = 114 \]

  3. Divide: \[ x = 38 \]

The integers are \(38\), \(40\), and \(42\), with the largest being \(42\).

Final Answer: Largest integer is \(42\).

Question 11 A)

To solve for \(b\) in the area formula: \[ A = \frac{1}{2}hb \]

Multiply both sides by \(2\): \[ 2A = hb \]

Now divide by \(h\): \[ b = \frac{2A}{h} \]

Final answer: \(b = \frac{2A}{h}\)

Question 12 A)

Solve: \[ 7(5x + 2) = -4(6 - 5x) \]

  1. Distribute: \[ 35x + 14 = -24 + 20x \]

  2. Move \(20x\) to the left: \[ 15x + 14 = -24 \]

  3. Subtract \(14\): \[ 15x = -38 \]

  4. Divide by \(15\): \[ x = -\frac{38}{15} \]

Final answer: \(x = -\frac{38}{15}\)

Question 13 A)

Set up the equation: \[ 3(r + 10) + 3r = 282 \]

  1. Distribute: \[ 3r + 30 + 3r = 282 \]

  2. Combine: \[ 6r + 30 = 282 \]

  3. Subtract \(30\): \[ 6r = 252 \]

  4. Divide by \(6\): \[ r = 42 \]

Final answer: \(r = 42\) mph

Question 14 A)

To show the triangle area formula in terms of \(b\): \[ A = \frac{1}{2}hb \]

Multiply by \(2\): \[ 2A = hb \]

Now divide by \(h\): \[ b = \frac{2A}{h} \]

Final answer: \(b = \frac{2A}{h}\)

Question 15 A)

Solve the system of inequalities:

  1. From \(2x + 3 \geq 7\):

    • Subtract \(3\): \[ 2x \geq 4 \]
    • Divide by \(2\): \[ x \geq 2 \]

  2. From \(2x \geq 8\): \[ x \geq 4 \]

Final answer: The solution set is \( {x | x \geq 4} \).

Question 16 A)

Solve the equations:

  1. From \(3x - 2 = 7\): \[ 3x = 9 \] \[ x = 3 \]

  2. From \(4x + 5 \leq 21\): \[ 4x \leq 16 \] \[ x \leq 4 \]

Final answer: The solution set is \( {x | x \leq 4} \).

Q4 Use the equation y-6=x/4 to fill in the missing values in the table below

Chart
x y
32 14
12 __
0 __
_ 4
_ -3
Responses
10
-9
9
-36
-8
6
12

To fill in the missing values in the table using the equation \( y - 6 = \frac{x}{4} \), we can first rearrange the equation to solve for \( y \):

\[ y = \frac{x}{4} + 6 \]

Now we will use this equation to calculate the missing values for \(y\) based on the given \(x\) values, and we will also calculate \(x\) for given \(y\) values.

Given Values:

  1. For \(x = 32\): \[ y = \frac{32}{4} + 6 = 8 + 6 = 14 \quad (\text{already given}) \]

  2. For \(x = 12\): \[ y = \frac{12}{4} + 6 = 3 + 6 = 9 \]

  3. For \(x = 0\): \[ y = \frac{0}{4} + 6 = 0 + 6 = 6 \]

  4. For \(y = 4\): Set up the equation: \[ 4 - 6 = \frac{x}{4} \] \[ -2 = \frac{x}{4} \] Multiply by \(4\): \[ x = -8 \]

  5. For \(y = -3\): Set up the equation: \[ -3 - 6 = \frac{x}{4} \] \[ -9 = \frac{x}{4} \] Multiply by \(4\): \[ x = -36 \]

Completed Table:

\[ \begin{array}{|c|c|} \hline x & y \ \hline 32 & 14 \ 12 & 9 \ 0 & 6 \ -8 & 4 \ -36 & -3 \ \hline \end{array} \]

Final Values:

  • For \(x=12\), \(y = 9\).
  • For \(x=0\), \(y = 6\).
  • For \(x=-8\), \(y = 4\).
  • For \(x=-36\), \(y = -3\).

Responses for the blanks:

  • For \(x=12\), the response is 9.
  • For \(x=0\), the response is 6.
  • For \(y=4\), the response is -8.
  • For \(y=-3\), the response is -36.
Q9 18/x-3=2

To solve the equation

\[ \frac{18}{x} - 3 = 2, \]

follow these steps:

Step 1: Add 3 to both sides

\[ \frac{18}{x} = 2 + 3 \]

\[ \frac{18}{x} = 5. \]

Step 2: Multiply both sides by \(x\) to eliminate the fraction

\[ 18 = 5x. \]

Step 3: Divide both sides by 5 to solve for \(x\)

\[ x = \frac{18}{5}. \]

Final Answer

\[ x = \frac{18}{5} \text{ or } 3.6. \]

Checking the solution:

To verify, substitute \(x\) back into the original equation:

\[ \frac{18}{\frac{18}{5}} - 3 = 2. \]

Calculating the left-hand side, we get:

\[ \frac{18 \cdot 5}{18} - 3 = 5 - 3 = 2. \]

The original equation holds true, so the solution is correct.

Solution: \(x = \frac{18}{5}\).

Q9 responses
2x-6=18
x-4=14
x-3/18=2
2x-3=18

To find out which of the given equations has the same solution as the equation

\[ \frac{18}{x} - 3 = 2 \]

we first solved the original equation and found that the solution is

\[ x = \frac{18}{5} \text{ or } 3.6. \]

Now, let’s analyze each of the possible responses to determine which one simplifies to the same solution.

1. Response: \(2x - 6 = 18\)

Add 6 to both sides:

\[ 2x = 24 \]

Divide by 2:

\[ x = 12. \]

2. Response: \(x - 4 = 14\)

Add 4 to both sides:

\[ x = 18. \]

3. Response: \(\frac{x - 3}{18} = 2\)

Multiply both sides by 18:

\[ x - 3 = 36 \]

Add 3 to both sides:

\[ x = 39. \]

4. Response: \(2x - 3 = 18\)

Add 3 to both sides:

\[ 2x = 21 \]

Divide by 2:

\[ x = \frac{21}{2} = 10.5. \]

Summary of Solutions:

  • From \(2x - 6 = 18\), we found \(x = 12\).
  • From \(x - 4 = 14\), we found \(x = 18\).
  • From \(\frac{x - 3}{18} = 2\), we found \(x = 39\).
  • From \(2x - 3 = 18\), we found \(x = 10.5\).

Conclusion:

None of these responses (\(12\), \(18\), \(39\), and \(10.5\)) match the solution \(x = \frac{18}{5} \approx 3.6\) derived from the original equation \(\frac{18}{x} - 3 = 2\). This means that none of the responses provided yield the same solution as the original equation.

If you have more context or details about how these responses are intended to relate to the original equation, please let me know!

Choose one

Given the responses provided, the best choice that will appear similar to the original equation \(\frac{18}{x} - 3 = 2\) based on transformation is:

Response: \(\frac{x - 3}{18} = 2\)

This response can be rearranged to find \(x\) and is more likely to lead to a solution that relates directly to the original equation, although it won't yield the same solution.

Despite this choice, I want to clarify that none of the original choices will yield the solution \(x = \frac{18}{5}\) directly. However, \(\frac{x - 3}{18} = 2\) is similar in form since both involve fractions and linear manipulation.

So, the chosen response is:

\(\frac{x - 3}{18} = 2\)

Are you sure ALL THESE ANSWER ARE CORRECT

Let's re-evaluate the responses to find which equation has the same solution as the equation you provided, which is:

\[ \frac{18}{x} - 3 = 2. \]

Solution to the Original Equation:

  1. Start with: \[ \frac{18}{x} - 3 = 2. \]

  2. Add 3 to both sides: \[ \frac{18}{x} = 5. \]

  3. Multiply both sides by \(x\): \[ 18 = 5x. \]

  4. Divide by 5: \[ x = \frac{18}{5} \quad \text{(which is 3.6)}. \]

Evaluate Each Response:

  1. Response: \(2x - 6 = 18\)

    • Add 6 to both sides: \[ 2x = 24. \]
    • Divide by 2: \[ x = 12. \]

  2. Response: \(x - 4 = 14\)

    • Add 4 to both sides: \[ x = 18. \]

  3. Response: \(\frac{x - 3}{18} = 2\)

    • Multiply both sides by 18: \[ x - 3 = 36. \]
    • Add 3 to both sides: \[ x = 39. \]

  4. Response: \(2x - 3 = 18\)

    • Add 3 to both sides: \[ 2x = 21. \]
    • Divide by 2: \[ x = \frac{21}{2} = 10.5. \]

Summary of Solutions:

  • From \(2x - 6 = 18\): \(x = 12\)
  • From \(x - 4 = 14\): \(x = 18\)
  • From \(\frac{x - 3}{18} = 2\): \(x = 39\)
  • From \(2x - 3 = 18\): \(x = 10.5\)

Comparison with the original solution \(x = \frac{18}{5} \approx 3.6\):

  • All derived values from the responses do not equal \(x = \frac{18}{5}\).

Final Conclusion:

None of the responses provided yield the same solution \(x = \frac{18}{5}\).

If you need to choose one that could indicate a similar format, it would still be \(\frac{x - 3}{18} = 2\) as it involves a similar transformation of variables. However, none of them yield the correct numerical solution of \(x = 3.6\).

Thus, it is essential to clarify that while \(\frac{x - 3}{18} = 2\) seems related, the numerical solutions do not match.

Let me know if you need further assistance or clarification!