Answer :
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]
### 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]
