Answer :
Let's carefully go through the solution to the inequality [tex]\(-4\left(\frac{5}{2} + \frac{3}{2} x \right) > 8\)[/tex] step-by-step to identify any errors Mark may have made and solve the inequality correctly.
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### Part A: Identify the Errors
Given the inequality [tex]\( -4\left(\frac{5}{2} + \frac{3}{2} x \right) > 8 \)[/tex],
#### Step 1: Distribute [tex]\(-4\)[/tex]
[tex]\[ -4 \left(\frac{5}{2}\right) + -4 \left(\frac{3}{2} x \right) \][/tex]
This becomes:
[tex]\[ -10 - 6x > 8 \][/tex]
This step is correctly simplified.
#### Step 2: Isolate the term with [tex]\(x\)[/tex] (Add 10 to both sides)
[tex]\[ -10 - 6x + 10 > 8 + 10 \][/tex]
Which simplifies to:
[tex]\[ -6x > 18 \][/tex]
This step correctly isolates the term with [tex]\(x\)[/tex].
#### Step 3: Divide by [tex]\(-6\)[/tex] and reverse the inequality sign
When you divide by a negative number, you must reverse the inequality sign:
[tex]\[ x < \frac{18}{-6} \][/tex]
Which simplifies to:
[tex]\[ x < -3 \][/tex]
This step is done correctly.
In summary, Mark's steps look fine but let’s check:
- Step 1: [tex]\(-10 - 6x > 8\)[/tex] (Correct, no error)
- Step 2: [tex]\(-6x > 18\)[/tex] (Correct, no error)
- Step 3: [tex]\(x < -3\)[/tex] (Correct, no error)
- None of the given steps (A, B, C, D) in Mark's work contain errors.
Hence, there are no steps showing an error.
### Part B: Correct Solution to the Original Inequality
Let's run through the correct steps to solve the given inequality:
[tex]\[ -4\left(\frac{5}{2} + \frac{3}{2} x \right) > 8 \][/tex]
#### 1. Distribute [tex]\(-4\)[/tex]:
[tex]\[ -4 \left(\frac{5}{2}\right) + -4 \left(\frac{3}{2} x \right) > 8 \][/tex]
This simplifies to:
[tex]\[ -10 - 6x > 8 \][/tex]
#### 2. Add 10 to both sides:
[tex]\[ -10 - 6x + 10 > 8 + 10 \][/tex]
Which simplifies to:
[tex]\[ -6x > 18 \][/tex]
#### 3. Divide by [tex]\(-6\)[/tex] and reverse the inequality sign:
[tex]\[ x < \frac{18}{-6} \][/tex]
Which simplifies to:
[tex]\[ x < -3 \][/tex]
So, the correct solution to the inequality is:
[tex]\[ x < -3 \][/tex]
In conclusion, Mark's solution had no errors, and the correct solution to the original inequality is [tex]\( \boxed{x < -3} \)[/tex].
---
### Part A: Identify the Errors
Given the inequality [tex]\( -4\left(\frac{5}{2} + \frac{3}{2} x \right) > 8 \)[/tex],
#### Step 1: Distribute [tex]\(-4\)[/tex]
[tex]\[ -4 \left(\frac{5}{2}\right) + -4 \left(\frac{3}{2} x \right) \][/tex]
This becomes:
[tex]\[ -10 - 6x > 8 \][/tex]
This step is correctly simplified.
#### Step 2: Isolate the term with [tex]\(x\)[/tex] (Add 10 to both sides)
[tex]\[ -10 - 6x + 10 > 8 + 10 \][/tex]
Which simplifies to:
[tex]\[ -6x > 18 \][/tex]
This step correctly isolates the term with [tex]\(x\)[/tex].
#### Step 3: Divide by [tex]\(-6\)[/tex] and reverse the inequality sign
When you divide by a negative number, you must reverse the inequality sign:
[tex]\[ x < \frac{18}{-6} \][/tex]
Which simplifies to:
[tex]\[ x < -3 \][/tex]
This step is done correctly.
In summary, Mark's steps look fine but let’s check:
- Step 1: [tex]\(-10 - 6x > 8\)[/tex] (Correct, no error)
- Step 2: [tex]\(-6x > 18\)[/tex] (Correct, no error)
- Step 3: [tex]\(x < -3\)[/tex] (Correct, no error)
- None of the given steps (A, B, C, D) in Mark's work contain errors.
Hence, there are no steps showing an error.
### Part B: Correct Solution to the Original Inequality
Let's run through the correct steps to solve the given inequality:
[tex]\[ -4\left(\frac{5}{2} + \frac{3}{2} x \right) > 8 \][/tex]
#### 1. Distribute [tex]\(-4\)[/tex]:
[tex]\[ -4 \left(\frac{5}{2}\right) + -4 \left(\frac{3}{2} x \right) > 8 \][/tex]
This simplifies to:
[tex]\[ -10 - 6x > 8 \][/tex]
#### 2. Add 10 to both sides:
[tex]\[ -10 - 6x + 10 > 8 + 10 \][/tex]
Which simplifies to:
[tex]\[ -6x > 18 \][/tex]
#### 3. Divide by [tex]\(-6\)[/tex] and reverse the inequality sign:
[tex]\[ x < \frac{18}{-6} \][/tex]
Which simplifies to:
[tex]\[ x < -3 \][/tex]
So, the correct solution to the inequality is:
[tex]\[ x < -3 \][/tex]
In conclusion, Mark's solution had no errors, and the correct solution to the original inequality is [tex]\( \boxed{x < -3} \)[/tex].