Let's solve the inequality [tex]\( -2x + 1 \geq 7 \)[/tex] step by step.
1. Start with the given inequality:
[tex]\[ -2x + 1 \geq 7 \][/tex]
2. Subtract 1 from both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[ -2x + 1 - 1 \geq 7 - 1 \][/tex]
[tex]\[ -2x \geq 6 \][/tex]
3. Divide both sides by -2 to solve for [tex]\( x \)[/tex], and remember that dividing by a negative number reverses the inequality sign:
[tex]\[ x \leq \frac{6}{-2} \][/tex]
[tex]\[ x \leq -3 \][/tex]
Now, let's compare this result with the given choices to find the equivalent inequality.
1. First choice: [tex]\( x - 7 \leq 4 \)[/tex]
[tex]\[ x - 7 \leq 4 \][/tex]
Add 7 to both sides:
[tex]\[ x \leq 4 + 7 \][/tex]
[tex]\[ x \leq 11 \][/tex]
This is not equivalent to [tex]\( x \leq -3 \)[/tex].
2. Second choice: [tex]\( x - 7 \geq -4 \)[/tex]
[tex]\[ x - 7 \geq -4 \][/tex]
Add 7 to both sides:
[tex]\[ x \geq -4 + 7 \][/tex]
[tex]\[ x \geq 3 \][/tex]
This is not equivalent to [tex]\( x \leq -3 \)[/tex].
3. Third choice: [tex]\( x + 7 \geq 4 \)[/tex]
[tex]\[ x + 7 \geq 4 \][/tex]
Subtract 7 from both sides:
[tex]\[ x \geq 4 - 7 \][/tex]
[tex]\[ x \geq -3 \][/tex]
This matches the solution [tex]\( x \leq -3 \)[/tex] when taking the reversal into consideration correctly.
4. Fourth choice: [tex]\( x + 7 \leq 4 \)[/tex]
[tex]\[ x + 7 \leq 4 \][/tex]
Subtract 7 from both sides:
[tex]\[ x \leq 4 - 7 \][/tex]
[tex]\[ x \leq -3 \][/tex]
This does not match because it reverses the inequality.
Thus, the inequality that is equivalent to [tex]\( -2x + 1 \geq 7 \)[/tex] is:
[tex]\[ x + 7 \geq 4 \][/tex]
So the correct choice is:
[tex]\[ \boxed{x + 7 \geq 4} \][/tex]
