To solve this problem, we need to address each inequality individually and then combine their solutions. Here are the steps:
### Step 1: Solve the first inequality [tex]\(4x + 2 > 14\)[/tex]
1. Subtract 2 from both sides of the inequality:
[tex]\[
4x + 2 - 2 > 14 - 2
\][/tex]
Simplifies to:
[tex]\[
4x > 12
\][/tex]
2. Divide both sides by 4:
[tex]\[
\frac{4x}{4} > \frac{12}{4}
\][/tex]
Simplifies to:
[tex]\[
x > 3
\][/tex]
### Step 2: Solve the second inequality [tex]\(-21x + 1 > 22\)[/tex]
1. Subtract 1 from both sides of the inequality:
[tex]\[
-21x + 1 - 1 > 22 - 1
\][/tex]
Simplifies to:
[tex]\[
-21x > 21
\][/tex]
2. Divide both sides by -21. Note that dividing by a negative number reverses the inequality sign:
[tex]\[
\frac{-21x}{-21} < \frac{21}{-21}
\][/tex]
Simplifies to:
[tex]\[
x < -1
\][/tex]
### Combining the solutions:
- From the first inequality, we have [tex]\(x > 3\)[/tex].
- From the second inequality, we have [tex]\(x < -1\)[/tex].
Thus, we need [tex]\(x\)[/tex] to be both greater than 3 and less than -1 simultaneously.
### Conclusion:
There are no real numbers [tex]\(x\)[/tex] that satisfy both inequalities at the same time. Therefore, the solution set is the empty set.
[tex]\[ \boxed{\text{No real numbers } x \text{ satisfy both inequalities}} \][/tex]
