To solve the compound inequality [tex]\( 2x - 4 < -10 \)[/tex] or [tex]\( 4x + 4 \geq 16 \)[/tex], we will handle each inequality separately and then combine the results.
### Step-by-Step Solution
#### Solving the first inequality [tex]\( 2x - 4 < -10 \)[/tex]
1. Isolate the variable [tex]\( x \)[/tex]:
[tex]\[
2x - 4 < -10
\][/tex]
2. Add 4 to both sides:
[tex]\[
2x - 4 + 4 < -10 + 4 \\
2x < -6
\][/tex]
3. Divide both sides by 2:
[tex]\[
x < \frac{-6}{2} \\
x < -3
\][/tex]
So, the solution to the first inequality is:
[tex]\[
x < -3
\][/tex]
#### Solving the second inequality [tex]\( 4x + 4 \geq 16 \)[/tex]
1. Isolate the variable [tex]\( x \)[/tex]:
[tex]\[
4x + 4 \geq 16
\][/tex]
2. Subtract 4 from both sides:
[tex]\[
4x + 4 - 4 \geq 16 - 4 \\
4x \geq 12
\][/tex]
3. Divide both sides by 4:
[tex]\[
x \geq \frac{12}{4} \\
x \geq 3
\][/tex]
So, the solution to the second inequality is:
[tex]\[
x \geq 3
\][/tex]
### Combine the Solutions
Since the original inequality is connected by "or," we combine the two sets of solutions. This means we are looking for [tex]\( x \)[/tex] values that satisfy either one or both of the inequalities. The combined solution is:
[tex]\[
x < -3 \quad \text{or} \quad x \geq 3
\][/tex]
### Graphing the Solution on the Number Line
1. Draw a number line.
2. For [tex]\( x < -3 \)[/tex]:
- Shade all values to the left of -3 (but do not include -3).
- Use an open circle at -3 to indicate that -3 is not included.
3. For [tex]\( x \geq 3 \)[/tex]:
- Shade all values to the right of and including 3.
- Use a closed circle at 3 to indicate that 3 is included.
The number line should look like this:
[tex]\[
\begin{array}{c}
\ \ \ \ \ \bullet \ \ \ \ \ \ \ \ \ \ \ \ \bullet\\
\ <<<<<|>>>>>>>> \quad \quad
\\ \quad -3 \quad \quad 3\\
\end{array}
\][/tex]
In summary, the combined solution to the inequality [tex]\( 2x - 4 < -10 \)[/tex] or [tex]\( 4x + 4 \geq 16 \)[/tex] is [tex]\( x < -3 \)[/tex] or [tex]\( x \geq 3 \)[/tex], and the graph on the number line represents these intervals.
