To solve the inequality [tex]\(7x + 5 < 19 - 3x\)[/tex], follow these steps:
1. Simplify the inequality by isolating the variable term on one side:
Begin by adding [tex]\(3x\)[/tex] to both sides of the inequality to get the variable term on one side.
[tex]\[
7x + 5 + 3x < 19 - 3x + 3x
\][/tex]
This simplifies to:
[tex]\[
10x + 5 < 19
\][/tex]
2. Isolate the variable term further by moving the constant term to the other side:
Subtract 5 from both sides of the inequality:
[tex]\[
10x + 5 - 5 < 19 - 5
\][/tex]
This simplifies to:
[tex]\[
10x < 14
\][/tex]
3. Solve for the variable [tex]\(x\)[/tex]:
Divide both sides of the inequality by 10 to isolate [tex]\(x\)[/tex]:
[tex]\[
\frac{10x}{10} < \frac{14}{10}
\][/tex]
This simplifies to:
[tex]\[
x < \frac{14}{10}
\][/tex]
Simplify the fraction:
[tex]\[
x < \frac{7}{5}
\][/tex]
So, the solution to the inequality [tex]\(7x + 5 < 19 - 3x\)[/tex] is:
[tex]\[
x < \frac{7}{5}
\][/tex]
Thus, the interval representing the solution set is:
[tex]\[
(-\infty, \frac{7}{5})
\][/tex]
