Answer :

Sure, let's solve the inequality [tex]\( |6x| + 8 \leq 20 \)[/tex] step by step.

### Step 1: Isolate the Absolute Value

We start by isolating the absolute value expression:

[tex]\[ |6x| + 8 \leq 20 \][/tex]

Subtract 8 from both sides:

[tex]\[ |6x| \leq 12 \][/tex]

### Step 2: Break Down the Absolute Value Inequality

The absolute value inequality [tex]\( |6x| \leq 12 \)[/tex] means that the expression inside the absolute value, [tex]\( 6x \)[/tex], lies within the interval from [tex]\(-12\)[/tex] to [tex]\(12\)[/tex]. Mathematically, this can be written as:

[tex]\[ -12 \leq 6x \leq 12 \][/tex]

### Step 3: Solve the Compound Inequality

Now, we need to solve the compound inequality for [tex]\( x \)[/tex]. We do this by dividing all parts of the inequality by 6:

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

### Step 4: Double-Check the Solution

Let's confirm by breaking down the absolute value into its two possible cases:

#### Case 1: [tex]\(6x \geq 0\)[/tex]

For [tex]\(6x\)[/tex] being non-negative:

[tex]\[ 6x \leq 12 \][/tex]

Divide by 6:

[tex]\[ x \leq 2 \][/tex]

#### Case 2: [tex]\(6x < 0\)[/tex]

For [tex]\(6x\)[/tex] being negative:

[tex]\[ -6x \leq 12 \][/tex]

Divide by -6 (and flip the inequality sign):

[tex]\[ x \geq -2 \][/tex]

### Step 5: Combine the Results

Combining both cases, we have the solution interval:

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

So, the solution to the inequality [tex]\( |6x| + 8 \leq 20 \)[/tex] is:

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

This means [tex]\( x \)[/tex] must lie within the interval [tex]\([-2, 2]\)[/tex].