To solve the compound inequality [tex]\(\frac{x}{4} + 7 < -1\)[/tex] or [tex]\(2x - 1 \geq 9\)[/tex], we need to address each part of the inequality separately and then combine the solutions.
Step 1: Solve [tex]\(\frac{x}{4} + 7 < -1\)[/tex]
1.1. Start by isolating [tex]\(x\)[/tex] on one side of the inequality:
[tex]\[
\frac{x}{4} + 7 < -1
\][/tex]
1.2. Subtract 7 from both sides:
[tex]\[
\frac{x}{4} < -1 - 7
\][/tex]
1.3. Simplify the right-hand side:
[tex]\[
\frac{x}{4} < -8
\][/tex]
1.4. Multiply both sides by 4 to solve for [tex]\(x\)[/tex]:
[tex]\[
x < -32
\][/tex]
The solution to the first part is all [tex]\(x\)[/tex] values less than [tex]\(-32\)[/tex].
Step 2: Solve [tex]\(2x - 1 \geq 9\)[/tex]
2.1. Start by isolating [tex]\(x\)[/tex] on one side of the inequality:
[tex]\[
2x - 1 \geq 9
\][/tex]
2.2. Add 1 to both sides:
[tex]\[
2x \geq 9 + 1
\][/tex]
2.3. Simplify the right-hand side:
[tex]\[
2x \geq 10
\][/tex]
2.4. Divide both sides by 2 to solve for [tex]\(x\)[/tex]:
[tex]\[
x \geq 5
\][/tex]
The solution to the second part is all [tex]\(x\)[/tex] values greater than or equal to 5.
Step 3: Combine the Solutions
Since the compound inequality uses "or," we need to find the union of the two solution sets:
- For the first inequality: [tex]\(x < -32\)[/tex]
- For the second inequality: [tex]\(x \geq 5\)[/tex]
The combined solution set includes all values of [tex]\(x\)[/tex] either less than [tex]\(-32\)[/tex] or greater than or equal to 5.
Final Solution:
The solution set can be written as:
[tex]\[
(-\infty, -32) \cup [5, \infty)
\][/tex]
This represents all values of [tex]\(x\)[/tex] that solve the given compound inequality.
