Sure, let's solve the inequality step-by-step.
The given inequality is:
[tex]\[ 17 > 3(2 + 4x) - (1 - 12x) \][/tex]
### Step 1: Distribute the constants inside the parentheses
Distributing the 3 inside the parentheses on the left-hand side:
[tex]\[ 3(2 + 4x) = 3 \cdot 2 + 3 \cdot 4x = 6 + 12x \][/tex]
Distributing the -1 inside the parentheses on the right-hand side:
[tex]\[ -(1 - 12x) = -1 + 12x \][/tex]
So, the inequality becomes:
[tex]\[ 17 > 6 + 12x - 1 + 12x \][/tex]
### Step 2: Combine like terms
Combine the constant terms and the terms involving [tex]\( x \)[/tex] on the right-hand side:
[tex]\[ 6 - 1 = 5 \][/tex]
[tex]\[ 12x + 12x = 24x \][/tex]
Therefore, the inequality simplifies to:
[tex]\[ 17 > 5 + 24x \][/tex]
### Step 3: Isolate the term involving [tex]\( x \)[/tex]
Move the constant term 5 to the left-hand side by subtracting 5 from both sides:
[tex]\[ 17 - 5 > 24x \][/tex]
[tex]\[ 12 > 24x \][/tex]
### Step 4: Solve for [tex]\( x \)[/tex]
Divide both sides by 24 to solve for [tex]\( x \)[/tex]:
[tex]\[ \frac{12}{24} > x \][/tex]
This simplifies to:
[tex]\[ \frac{1}{2} > x \][/tex]
### Step 5: Write the simplified inequality
The inequality can also be written as:
[tex]\[ x < \frac{1}{2} \][/tex]
So, the solution to the inequality [tex]\( 17 > 3(2 + 4x) - (1 - 12x) \)[/tex] is [tex]\( x < \frac{1}{2} \)[/tex].
