The Correct Answer and Explanation is:
Of course. Here is the correct answer and a detailed explanation.
Correct Answer
Inequality Solution: -20 < x < -8
Number Line: The graph should show an open circle at -20 and another open circle at -8, with the line segment between these two points shaded.
Explanation
The problem asks us to solve the compound inequality -10 > x – 2 > -22. A compound inequality like this one involves three parts: a left side, a middle, and a right side. Our goal is to isolate the variable ‘x’ in the middle part. The key principle is that whatever operation we perform to the middle part, we must also perform to the left and right parts to keep the inequality true.
First, we identify the operation being done to ‘x’ in the middle section, which is ‘x – 2’. To isolate ‘x’, we need to perform the inverse operation of subtracting 2, which is adding 2. We will add 2 to all three parts of the inequality.
The original inequality is:
-10 > x – 2 > -22
Now, we add 2 to each part:
-10 + 2 > (x – 2) + 2 > -22 + 2
Calculating the result for each part gives us:
-8 > x > -20
This is a correct solution, but it is standard mathematical convention to write inequalities with the smaller number on the left and the larger number on the right. This makes it easier to read and graph. To rewrite it, we can simply flip the entire statement, which also reverses the direction of the inequality signs.
So, -8 > x > -20 is equivalent to:
-20 < x < -8
This final form tells us that ‘x’ can be any number that is greater than -20 but also less than -8.
To graph this solution on a number line, we place circles at the two endpoints, -20 and -8. Since the inequality symbols are < (less than) and not ≤ (less than or equal to), the endpoints themselves are not included in the solution set. We represent this by using open circles. Finally, we shade the region on the number line between the two open circles, representing all the possible values for ‘x’.
