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].
