Answered

IDNLearn.com: Your trusted platform for finding precise and reliable answers. Ask your questions and receive reliable and comprehensive answers from our dedicated community of professionals.

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.

---

### 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].
Thank you for using this platform to share and learn. Don't hesitate to keep asking and answering. We value every contribution you make. IDNLearn.com has the answers you need. Thank you for visiting, and we look forward to helping you again soon.