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Mark incorrectly solved the inequality [tex]-4\left(\frac{5}{2}+\frac{3}{2} x\right)\ \textgreater \ 8[/tex]. His work is shown.

Part A: Which step shows an error based on the inequality only from the previous step? Select all that apply.

A. Step 1: [tex]-10 + 6x \ \textgreater \ 8[/tex]

B. Step 2: [tex]6x \ \textgreater \ 8 - 10[/tex]

C. Step 3: [tex]6x \ \textgreater \ -2[/tex]

D. Step 4: [tex]x \ \textgreater \ -\frac{1}{3}[/tex]

Part B: What is the correct solution to the original inequality?

[tex]\qquad[/tex]

Sagot :

GinnyAnsweridn
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].
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