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What is the solution to [tex]|2x - 8| \leq 6[/tex]?

A. [tex]-7 \leq x \leq -1[/tex]
B. [tex]1 \leq x \leq 7[/tex]
C. [tex]x \leq 1[/tex] or [tex]x \geq 7[/tex]
D. [tex]x \leq -7[/tex] or [tex]x \geq -1[/tex]

Sagot :

GinnyAnsweridn
To solve the inequality [tex]\( |2x - 8| \leq 6 \)[/tex], you need to break it into two separate inequalities and solve them individually. The absolute value inequality [tex]\( |A| \leq B \)[/tex] can be rewritten as:

[tex]\[ -B \leq A \leq B \][/tex]

In our specific problem, [tex]\( A = 2x - 8 \)[/tex] and [tex]\( B = 6 \)[/tex], so we have:

[tex]\[ -6 \leq 2x - 8 \leq 6 \][/tex]

We'll solve this compound inequality in two steps.

### Step 1: Solve [tex]\( -6 \leq 2x - 8 \)[/tex]
1. Add 8 to both sides:
[tex]\[ -6 + 8 \leq 2x \][/tex]
[tex]\[ 2 \leq 2x \][/tex]

2. Divide both sides by 2:
[tex]\[ 1 \leq x \][/tex]
[tex]\[ x \geq 1 \][/tex]

### Step 2: Solve [tex]\( 2x - 8 \leq 6 \)[/tex]
1. Add 8 to both sides:
[tex]\[ 2x - 8 + 8 \leq 6 + 8 \][/tex]
[tex]\[ 2x \leq 14 \][/tex]

2. Divide both sides by 2:
[tex]\[ x \leq 7 \][/tex]

### Combine the solutions:
The solution from Step 1 is [tex]\( x \geq 1 \)[/tex] and the solution from Step 2 is [tex]\( x \leq 7 \)[/tex].

Combining these, we get:
[tex]\[ 1 \leq x \leq 7 \][/tex]

Thus, the correct answer is:
B. [tex]\( 1 \leq x \leq 7 \)[/tex]
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