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Which of the following is the solution to the compound inequality below?
[tex]\[
5 + x \geq 3 \text{ or } 6x + 1 \ \textless \ -29
\][/tex]
A. [tex]\( x \geq -2 \)[/tex] or [tex]\( x \ \textless \ -5 \)[/tex] B. [tex]\( x \geq 8 \)[/tex] or [tex]\( x \ \textless \ \frac{14}{3} \)[/tex] C. [tex]\( x \geq -\frac{3}{5} \)[/tex] or [tex]\( x \ \textless \ -\frac{14}{3} \)[/tex] D. [tex]\( x \geq 2 \)[/tex] or [tex]\( x \ \textless \ 5 \)[/tex]
To determine the solution to the compound inequality [tex]\( 5 + x \geq 3 \)[/tex] or [tex]\( 6x + 1 < -29 \)[/tex], let's break it down into two parts and solve each part separately.
### Solving the first part: [tex]\( 5 + x \geq 3 \)[/tex]
1. Start with the inequality: [tex]\[ 5 + x \geq 3 \][/tex]
2. Subtract 5 from both sides to isolate [tex]\( x \)[/tex]: [tex]\[ x \geq 3 - 5 \][/tex]
3. Simplify: [tex]\[ x \geq -2 \][/tex]
### Solving the second part: [tex]\( 6x + 1 < -29 \)[/tex]
1. Start with the inequality: [tex]\[ 6x + 1 < -29 \][/tex]
2. Subtract 1 from both sides to isolate the term with [tex]\( x \)[/tex]: [tex]\[ 6x < -29 - 1 \][/tex]
3. Simplify: [tex]\[ 6x < -30 \][/tex]
4. Divide both sides by 6 to solve for [tex]\( x \)[/tex]: [tex]\[ x < -30 / 6 \][/tex]
5. Simplify: [tex]\[ x < -5 \][/tex]
### Combining the Solutions
The solution to the compound inequality [tex]\( 5 + x \geq 3 \)[/tex] or [tex]\( 6x + 1 < -29 \)[/tex] is the union of the solutions to each individual inequality.
- From the first part: [tex]\( x \geq -2 \)[/tex] - From the second part: [tex]\( x < -5 \)[/tex]
Thus, the combined solution is: [tex]\[ x \geq -2 \text{ or } x < -5 \][/tex]
Therefore, the correct choice is: [tex]\[ A. \, x \geq -2 \text{ or } x < -5 \][/tex]
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