To solve the inequality [tex]\( x - 4 > \frac{x}{2} + 8 \)[/tex], follow these steps:
1. Isolate the variable [tex]\( x \)[/tex]:
We start by getting all terms involving [tex]\( x \)[/tex] on one side and the constant terms on the other side. First, subtract [tex]\( \frac{x}{2} \)[/tex] from both sides to combine the [tex]\( x \)[/tex] terms:
[tex]\[ x - 4 - \frac{x}{2} > \frac{x}{2} + 8 - \frac{x}{2} \][/tex]
2. Simplify the inequality:
Combine like terms on the left side:
[tex]\[ x - \frac{x}{2} - 4 > 8 \][/tex]
Since [tex]\( x - \frac{x}{2} \)[/tex] is the same as [tex]\( \frac{2x}{2} - \frac{x}{2} \)[/tex], we can simplify this to [tex]\( \frac{2x - x}{2} = \frac{x}{2} \)[/tex]. So, the inequality becomes:
[tex]\[ \frac{x}{2} - 4 > 8 \][/tex]
3. Isolate [tex]\( \frac{x}{2} \)[/tex]:
Add 4 to both sides to isolate the term involving [tex]\( x \)[/tex]:
[tex]\[ \frac{x}{2} - 4 + 4 > 8 + 4 \][/tex]
This simplifies to:
[tex]\[ \frac{x}{2} > 12 \][/tex]
4. Solve for [tex]\( x \)[/tex]:
Multiply both sides of the inequality by 2 to solve for [tex]\( x \)[/tex]:
[tex]\[ 2 \times \frac{x}{2} > 12 \times 2 \][/tex]
This simplifies to:
[tex]\[ x > 24 \][/tex]
Therefore, the inequality [tex]\( x - 4 > \frac{x}{2} + 8 \)[/tex] is true for all values of [tex]\( x \)[/tex] such that [tex]\( x > 24 \)[/tex].
Thus, any value of [tex]\( x \)[/tex] that is greater than 24 is a solution to the inequality.
