The solution to the inequality `|10-2x|gt6` is

Options

A. `2xltxlt8`
B. `xlt-2` and `xgt8`
C. `xgt2` and `xlt-8`
D. `xlt2` or `xgt8`

Answer

To solve the inequality \( |10 - 2x| > 6 \), we will follow these steps:

### Step 1: Understand the Absolute Value Inequality
The expression \( |A| > B \) can be split into two cases:
1. \( A > B \)
2. \( A < -B \)

In our case, \( A = 10 - 2x \) and \( B = 6 \). Therefore, we can write:
1. \( 10 - 2x > 6 \)
2. \( 10 - 2x < -6 \)

### Step 2: Solve the First Inequality
Now, let's solve the first inequality:
\[ 10 - 2x > 6 \]
Subtract 10 from both sides:
\[ -2x > 6 - 10 \]
\[ -2x > -4 \]
Now, divide both sides by -2. Remember that when we divide by a negative number, we must reverse the inequality:
\[ x < 2 \]

### Step 3: Solve the Second Inequality
Now, let's solve the second inequality:
\[ 10 - 2x < -6 \]
Subtract 10 from both sides:
\[ -2x < -6 - 10 \]
\[ -2x < -16 \]
Now, divide both sides by -2, reversing the inequality:
\[ x > 8 \]

### Step 4: Combine the Solutions
From the two inequalities, we have:
1. \( x < 2 \)
2. \( x > 8 \)

Thus, the solution to the inequality \( |10 - 2x| > 6 \) is:
\[ x < 2 \quad \text{or} \quad x > 8 \]

### Final Answer
The solution to the inequality \( |10 - 2x| > 6 \) is \( x < 2 \) or \( x > 8 \).

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