Question

When solving compound inequalities whenever you solve for X do do you have to make X a positive if it’s a negative or do you leave as a negative

Answer 1

When solving compound inequalities, you do not need to make \( x \) positive if it is negative. Instead, you should isolate \( x \) based on the operations you are performing, keeping its sign according to the steps of your solution.

In a compound inequality, you generally follow these steps:

1. **Isolate the variable**: Perform the same operations on all parts of the inequality to isolate \( x \). This may involve adding, subtracting, multiplying, or dividing by numbers.

2. **Pay attention to signs when multiplying/dividing**: If you multiply or divide by a negative number, you must flip the inequality sign.

3. **Keep the sign of \( x \)**: If \( x \) ends up negative after the operations, you leave it as negative. The solution should reflect the actual values that satisfy the inequality.

**Example:**
Given the compound inequality \(-3 < 2x + 1 < 5\):

1. First, isolate \( 2x \):
\[
-4 < 2x < 4
\]

2. Next, divide each part by 2 to solve for \( x \):
\[
-2 < x < 2
\]

In this case, \( x \) can be negative (i.e., it is between -2 and 2), and you leave it as is.

So, in summary, you do not change the sign of \( x \) to make it positive; you solve for \( x \) according to the operations performed.