Let's analyze the given statement [tex]\( P: x + 9 = 10 \)[/tex] to determine which of the provided options is an equivalent statement.
Step 1: Solve the statement [tex]\( P \)[/tex].
Start with the equation:
[tex]\[ x + 9 = 10 \][/tex]
To isolate [tex]\( x \)[/tex], subtract 9 from both sides:
[tex]\[ x + 9 - 9 = 10 - 9 \][/tex]
[tex]\[ x = 1 \][/tex]
Step 2: Compare each option to [tex]\( P \)[/tex].
A) [tex]\( S: x = 1 \)[/tex]
This statement is equivalent to [tex]\( P \)[/tex] because solving [tex]\( P \)[/tex] gives [tex]\( x = 1 \)[/tex], which matches exactly.
B) [tex]\( R: x > 0 \)[/tex]
While [tex]\( x = 1 \)[/tex] satisfies this inequality, [tex]\( P \)[/tex] specifically states [tex]\( x = 1 \)[/tex]. The inequality [tex]\( x > 0 \)[/tex] suggests a range of possible values for [tex]\( x \)[/tex], which is broader than [tex]\( P \)[/tex]. Therefore, [tex]\( R \)[/tex] is not strictly equivalent to [tex]\( P \)[/tex].
C) [tex]\( \sim P: x + 9 \neq 10 \)[/tex]
This is the negation of [tex]\( P \)[/tex], which means it directly opposes the original statement. Therefore, it is not equivalent to [tex]\( P \)[/tex].
D) [tex]\( Q: x - 9 = 10 \)[/tex]
Solving this equation:
[tex]\[ x - 9 = 10 \][/tex]
Add 9 to both sides:
[tex]\[ x - 9 + 9 = 10 + 9 \][/tex]
[tex]\[ x = 19 \][/tex]
The value [tex]\( x = 19 \)[/tex] does not match the solution of [tex]\( P \)[/tex], so [tex]\( Q \)[/tex] is not equivalent to [tex]\( P \)[/tex].
The only option that is equivalent to [tex]\( P \)[/tex] is:
A) [tex]\( S: x = 1 \)[/tex]
