To solve the inequality [tex]\(6x + 5 + 2(4x + 1) > 6x + 9\)[/tex], follow these steps:
1. Expand and Simplify the Left Side:
[tex]\[
6x + 5 + 2(4x + 1)
\][/tex]
First, distribute the 2 inside the parentheses:
[tex]\[
2(4x + 1) = 8x + 2
\][/tex]
Now, substitute this back into the inequality:
[tex]\[
6x + 5 + 8x + 2 > 6x + 9
\][/tex]
2. Combine Like Terms on the Left Side:
[tex]\[
6x + 8x + 5 + 2 > 6x + 9
\][/tex]
Simplify:
[tex]\[
14x + 7 > 6x + 9
\][/tex]
3. Isolate the variable [tex]\(x\)[/tex]:
First, subtract [tex]\(6x\)[/tex] from both sides to remove the [tex]\(x\)[/tex] term from the right side:
[tex]\[
14x - 6x + 7 > 6x - 6x + 9
\][/tex]
Simplify:
[tex]\[
8x + 7 > 9
\][/tex]
4. Subtract 7 from Both Sides:
[tex]\[
8x + 7 - 7 > 9 - 7
\][/tex]
Simplify:
[tex]\[
8x > 2
\][/tex]
5. Divide by 8:
[tex]\[
\frac{8x}{8} > \frac{2}{8}
\][/tex]
Simplify:
[tex]\[
x > \frac{1}{4}
\][/tex]
Thus, the solution to the inequality [tex]\(6x + 5 + 2(4x + 1) > 6x + 9\)[/tex] is:
[tex]\[
x > \frac{1}{4}
\][/tex]
In interval notation, this solution can be expressed as:
[tex]\[
\left( \frac{1}{4}, \infty \right)
\][/tex]
