Let’s analyze the steps to identify the error in the inequality that Vera claimed.
First, let's rewrite the inequality Vera stated:
[tex]\[
-78.9999 \geq x
\][/tex]
This inequality means that [tex]\( x \)[/tex] can be any value less than or equal to [tex]\(-78.9999\)[/tex]. Now, let’s go through each given option to identify the error:
1. Vera wrote the inequality with the variable on the left side of the relation symbol.
[tex]\[
x \leq -78.9999
\][/tex]
It is perfectly valid to write the variable on either side of the inequality. The inequality is still correctly describing the same set of solutions. Thus, this is not an error.
2. Vera wrote a relation symbol that does not represent the direction of the ray.
[tex]\[
-78.9999 \geq x \Rightarrow x \leq -78.9999
\][/tex]
This inequality represents all numbers less than or equal to [tex]\(-78.9999\)[/tex]. The direction of the ray is correct because it includes all values to the left of [tex]\(-78.9999\)[/tex]. Therefore, this is not an error.
3. Vera selected an inequality that does not include -79 in its solution set.
[tex]\[
-79 \leq -78.9999
\][/tex]
In the inequality [tex]\(-78.9999 \ge x\)[/tex] ([tex]\( x \leq -78.9999\)[/tex]), [tex]\(-79\)[/tex] is indeed part of the solution set because [tex]\(-79\)[/tex] is less than [tex]\(-78.9999\)[/tex]. Hence, this is not an error.
4. Vera used the wrong number in her inequality.
The correct inequality should be [tex]\(-79 \geq x\)[/tex] (which means [tex]\( x \leq -79\)[/tex]) to precisely include [tex]\(-79\)[/tex] in the solution. By stating [tex]\(-78.9999 \geq x\)[/tex], she missed including the exact value [tex]\(-79\)[/tex] completely because [tex]\(-79\)[/tex] is not equal to [tex]\(-78.9999\)[/tex]. Thus, Vera should have used [tex]\(-79\)[/tex] instead of [tex]\(-78.9999\)[/tex].
Therefore, the correct error that Vera made is:
[tex]\[
\boxed{4}
\][/tex]
