To solve the inequality [tex]\( 4 - x < 2x - 5 \)[/tex], follow these steps:
1. Isolate the variable [tex]\( x \)[/tex] on one side of the inequality:
Start by eliminating [tex]\( x \)[/tex] from the left side. To do this, add [tex]\( x \)[/tex] to both sides of the inequality:
[tex]\[
4 - x + x < 2x - 5 + x
\][/tex]
Simplifying this, we get:
[tex]\[
4 < 3x - 5
\][/tex]
2. Isolate the constant term on one side:
Next, we need to eliminate the constant term on the right side. Add 5 to both sides of the inequality to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
4 + 5 < 3x - 5 + 5
\][/tex]
Simplifying this, we get:
[tex]\[
9 < 3x
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
To isolate [tex]\( x \)[/tex], divide both sides of the inequality by 3:
[tex]\[
\frac{9}{3} < \frac{3x}{3}
\][/tex]
Simplifying this, we get:
[tex]\[
3 < x
\][/tex]
4. Express the solution:
We have now found that [tex]\( x \)[/tex] must be greater than 3. In interval notation, this is written as:
[tex]\[
x \in (3, \infty)
\][/tex]
Thus, the solution to the inequality [tex]\( 4 - x < 2x - 5 \)[/tex] is:
[tex]\[
(3 < x) \text{ or } x \in (3, \infty).
\][/tex]
