Question

Solve for x.
x+12–5x>20 or x+2+11x>14
Write your answer as a compound inequality with integers.

Answer 1

To solve the inequalities separately, we first simplify them.

1. For the first inequality:
\[
x + 12 - 5x > 20
\]
Combine like terms:
\[
-4x + 12 > 20
\]
Subtract 12 from both sides:
\[
-4x > 8
\]
Divide both sides by -4 (this reverses the inequality sign):
\[
x < -2
\]

2. For the second inequality:
\[
x + 2 + 11x > 14
\]
Combine like terms:
\[
12x + 2 > 14
\]
Subtract 2 from both sides:
\[
12x > 12
\]
Divide both sides by 12:
\[
x > 1
\]

Now we combine the results of the two inequalities:
\[
x < -2 \quad \text{or} \quad x > 1
\]

The solution can be expressed as a compound inequality:
\[
(-\infty, -2) \cup (1, \infty)
\]

In integer terms, the solution can be written as:
\[
x < -2 \quad \text{or} \quad x > 1
\]
So the integers satisfying this are:
\[
\{ \ldots, -4, -3 \} \cup \{ 2, 3, 4, \ldots \}
\]

Thus, the final solution in integer form is \( x < -2 \) or \( x > 1 \).