To solve the inequality [tex]\( 9x - 4x + 8 < 3(x + 4) \)[/tex], we will work through it step-by-step:
1. Simplify both sides of the inequality:
Combine like terms on the left side of the inequality:
[tex]\[
9x - 4x + 8 < 3(x + 4)
\][/tex]
This simplifies to:
[tex]\[
5x + 8 < 3(x + 4)
\][/tex]
2. Distribute on the right side:
Distribute the 3 on the right side:
[tex]\[
5x + 8 < 3x + 12
\][/tex]
3. Isolate the variable term on one side:
Subtract [tex]\( 3x \)[/tex] from both sides to get the variable [tex]\( x \)[/tex] on one side:
[tex]\[
5x + 8 - 3x < 3x + 12 - 3x
\][/tex]
Simplify:
[tex]\[
2x + 8 < 12
\][/tex]
4. Isolate the variable further:
Subtract 8 from both sides to move the constant term to the right:
[tex]\[
2x + 8 - 8 < 12 - 8
\][/tex]
Simplify:
[tex]\[
2x < 4
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
Divide both sides by 2 to solve for [tex]\( x \)[/tex]:
[tex]\[
\frac{2x}{2} < \frac{4}{2}
\][/tex]
Simplify:
[tex]\[
x < 2
\][/tex]
The solution to the inequality [tex]\( 9x - 4x + 8 < 3(x + 4) \)[/tex] is [tex]\( x < 2 \)[/tex].
6. Write the solution set in interval notation:
In interval notation, the solution set is:
[tex]\[
(-\infty, 2)
\][/tex]
So, the complete solution in interval notation is [tex]\( (-\infty, 2) \)[/tex].
