To solve the inequality [tex]\( 11x - 6 < -18 + 5x \)[/tex], we will follow these steps:
1. Simplify the Inequality:
First, we need to collect all the [tex]\(x\)[/tex] terms on one side and the constants on the other side.
[tex]\[
11x - 6 < -18 + 5x
\][/tex]
2. Move [tex]\(5x\)[/tex] to the Left Side:
Subtract [tex]\(5x\)[/tex] from both sides to get the [tex]\(x\)[/tex] terms on one side.
[tex]\[
11x - 5x - 6 < -18
\][/tex]
Simplifying this, we get:
[tex]\[
6x - 6 < -18
\][/tex]
3. Move the Constant Term to the Right Side:
Add 6 to both sides to isolate the term with [tex]\(x\)[/tex] on the left side.
[tex]\[
6x - 6 + 6 < -18 + 6
\][/tex]
Simplifying this, we get:
[tex]\[
6x < -12
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Divide both sides by 6 to solve for [tex]\(x\)[/tex].
[tex]\[
x < -2
\][/tex]
Therefore, the solution to the inequality [tex]\( 11x - 6 < -18 + 5x \)[/tex] is [tex]\( x < -2 \)[/tex].
5. Write the Solution in Interval Notation:
The solution in interval notation is:
[tex]\[
(-\infty, -2)
\][/tex]
This interval represents all real numbers [tex]\(x\)[/tex] such that [tex]\(x\)[/tex] is less than [tex]\(-2\)[/tex].
