Let's solve the compound inequality step by step:
Given compound inequality:
[tex]\[ x + 7 < 3 \text{ or } x - 5 \geq -1 \][/tex]
We will solve each inequality separately.
### Solving the first part: [tex]\( x + 7 < 3 \)[/tex]
1. Isolate [tex]\( x \)[/tex] by subtracting 7 from both sides:
[tex]\[ x + 7 - 7 < 3 - 7 \][/tex]
2. Simplify the inequality:
[tex]\[ x < -4 \][/tex]
### Solving the second part: [tex]\( x - 5 \geq -1 \)[/tex]
1. Isolate [tex]\( x \)[/tex] by adding 5 to both sides:
[tex]\[ x - 5 + 5 \geq -1 + 5 \][/tex]
2. Simplify the inequality:
[tex]\[ x \geq 4 \][/tex]
### Combining the solutions
The solutions to the inequalities are:
- [tex]\( x < -4 \)[/tex]
- [tex]\( x \geq 4 \)[/tex]
These two inequalities are combined with the logical "or". This means [tex]\( x \)[/tex] satisfies either one of the inequalities. So the final solution is:
[tex]\[ x < -4 \text{ or } x \geq 4 \][/tex]
### Identifying the correct solution
The correct complete solution is:
[tex]\[ x < -4 \text{ or } x \geq 4 \][/tex]
From the provided options, we identify that:
A. Solution: [tex]\( x < -4 \text{ or } x \geq 4 \)[/tex]
B. Solution: [tex]\( x < -4 \text{ or } x \geq 4 \)[/tex]
C. Solution: [tex]\( x > -4 \text{ and } x \leq 4 \)[/tex]
D. Solution: [tex]\( x \leq -4 \text{ or } x > 4 \)[/tex]
Both options A and B present the correct solution notation. C and D are incorrect.
Thus, the correct solutions are:
A. Solution: [tex]\( x < -4 \text{ or } x \geq 4 \)[/tex]
and
B. Solution: [tex]\( x < -4 \text{ or } x \geq 4 \)[/tex]
So, either option A or B is correct.
