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Solve [tex]|2x - 4| \leq 8[/tex]

A. [tex]x \geq -3[/tex] or [tex]x \leq 2[/tex]

B. [tex]x \geq -2[/tex] and [tex]x \leq 6[/tex]

C. [tex]x \geq -2[/tex] and [tex]x \leq 4[/tex]

D. [tex]x \geq -2[/tex] and [tex]x \leq 5[/tex]


Sagot :

To solve the inequality [tex]\( |2x - 4| \leq 8 \)[/tex], we need to consider the definition of absolute value, which states that [tex]\( |a| \leq b \)[/tex] if and only if [tex]\( -b \leq a \leq b \)[/tex].

Given [tex]\( |2x - 4| \leq 8 \)[/tex], we can write this as:
[tex]\[ -8 \leq 2x - 4 \leq 8 \][/tex]

We will solve this compound inequality in two steps.

### Step 1: Solving the left inequality
[tex]\[ -8 \leq 2x - 4 \][/tex]

Add 4 to both sides:
[tex]\[ -8 + 4 \leq 2x - 4 + 4 \][/tex]
[tex]\[ -4 \leq 2x \][/tex]

Divide both sides by 2:
[tex]\[ -\frac{4}{2} \leq \frac{2x}{2} \][/tex]
[tex]\[ -2 \leq x \][/tex]

### Step 2: Solving the right inequality
[tex]\[ 2x - 4 \leq 8 \][/tex]

Add 4 to both sides:
[tex]\[ 2x - 4 + 4 \leq 8 + 4 \][/tex]
[tex]\[ 2x \leq 12 \][/tex]

Divide both sides by 2:
[tex]\[ \frac{2x}{2} \leq \frac{12}{2} \][/tex]
[tex]\[ x \leq 6 \][/tex]

### Combining the results
Combining the two parts of the compound inequality, we get:
[tex]\[ -2 \leq x \leq 6 \][/tex]

Therefore, the solution to [tex]\( |2x - 4| \leq 8 \)[/tex] is:
[tex]\[ x \geq -2 \text{ and } x \leq 6 \][/tex]

This matches option B.

So, the answer is:
B. [tex]\( x \geq -2 \text{ and } x \leq 6 \)[/tex]
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