Let's solve the compound inequality step-by-step:
We need to solve [tex]\(6x + 2 > 14\)[/tex] or [tex]\(3x - 1 < 2\)[/tex].
### Solving the First Inequality:
[tex]\[ 6x + 2 > 14 \][/tex]
1. Subtract 2 from both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[ 6x > 14 - 2 \][/tex]
[tex]\[ 6x > 12 \][/tex]
2. Divide both sides by 6 to solve for [tex]\(x\)[/tex]:
[tex]\[ x > \frac{12}{6} \][/tex]
[tex]\[ x > 2 \][/tex]
So, the solution to the first inequality is [tex]\( x > 2 \)[/tex].
### Solving the Second Inequality:
[tex]\[ 3x - 1 < 2 \][/tex]
1. Add 1 to both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[ 3x < 2 + 1 \][/tex]
[tex]\[ 3x < 3 \][/tex]
2. Divide both sides by 3 to solve for [tex]\(x\)[/tex]:
[tex]\[ x < \frac{3}{3} \][/tex]
[tex]\[ x < 1 \][/tex]
So, the solution to the second inequality is [tex]\( x < 1 \)[/tex].
### Combining Both Solutions:
Since we are dealing with an "or" compound inequality, we find the union of the solutions:
[tex]\[ x > 2 \quad \text{or} \quad x < 1 \][/tex]
Thus, the correct solution to the given compound inequality is [tex]\( x > 2 \)[/tex] or [tex]\( x < 1 \)[/tex].
### Choosing the Correct Option:
Looking at the given choices:
a. [tex]\( x > -2 \)[/tex] and [tex]\( x < -1 \)[/tex] (not matching)
b. [tex]\( x > \frac{16}{6} \)[/tex] or [tex]\( x < \frac{1}{3} \)[/tex] (not matching)
c. [tex]\( x > 2 \)[/tex] or [tex]\( x < 1 \)[/tex] (matches)
d. [tex]\( -2 < x > 1 \)[/tex] (not correct notation)
The correct answer is:
c. [tex]\( x > 2 \)[/tex] or [tex]\( x < 1 \)[/tex]
