To solve the inequality [tex]\(-x + 4 < \frac{1}{2} x + 1\)[/tex], follow these steps:
1. Isolate the variable: Start by getting all terms involving [tex]\( x \)[/tex] on one side of the inequality and the constants on the other side.
2. Combine like terms: Simplify the inequality by combining like terms.
Here's how we do it:
### Step-by-Step Solution:
1. Move all [tex]\( x \)[/tex]-terms to one side and constant terms to the other side:
[tex]\[
-x + 4 < \frac{1}{2} x + 1
\][/tex]
Subtract [tex]\(\frac{1}{2}x\)[/tex] from both sides to get all [tex]\( x \)[/tex]-terms on one side:
[tex]\[
-x - \frac{1}{2}x + 4 < 1
\][/tex]
2. Combine like terms involving [tex]\( x \)[/tex]:
[tex]\[
-\frac{3}{2}x + 4 < 1
\][/tex]
3. Isolate the [tex]\( x \)[/tex]-term:
Subtract 4 from both sides:
[tex]\[
-\frac{3}{2}x < 1 - 4
\][/tex]
[tex]\[
-\frac{3}{2}x < -3
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
To isolate [tex]\( x \)[/tex], divide both sides by [tex]\(-\frac{3}{2}\)[/tex]. Remember that dividing by a negative number reverses the inequality sign:
[tex]\[
x > \frac{-3}{-\frac{3}{2}}
\][/tex]
Simplify the right side:
[tex]\[
x > 2
\][/tex]
So the solution to the inequality is:
[tex]\[
x > 2
\][/tex]
### Conclusion:
The set of values for [tex]\( x \)[/tex] that satisfy the inequality [tex]\(-x + 4 < \frac{1}{2} x + 1\)[/tex] is [tex]\( x > 2 \)[/tex].
In interval notation, the solution is:
[tex]\[
x \in (2, \infty)
\][/tex]
This means that any value of [tex]\( x \)[/tex] greater than 2 will satisfy the given inequality.
