To solve the absolute value inequality |2x – 4| < 14, we start by breaking it down into two separate inequalities. The absolute value inequality |a| < b means that -b < a < b. In this case, a is 2x – 4 and b is 14.
So, we can rewrite the inequality as:
- -14 < 2x - 4 < 14
Now we will solve each part of the inequality:
- For the left part:
- -14 < 2x - 4
- Add 4 to both sides:
- -10 < 2x
- Now divide both sides by 2:
- -5 < x
- For the right part:
- 2x – 4 < 14
- Add 4 to both sides:
- 2x < 18
- Now divide both sides by 2:
- x < 9
Now we combine the two results. We have:
- -5 < x < 9
This indicates that x must be greater than -5 and less than 9.
Now, let’s graph this inequality on a number line:
- Draw a number line.
- Mark the points -5 and 9.
- Since we have an open inequality (<), use open circles at -5 and 9 to indicate that these points are not included in the solution.
- Shade the region between -5 and 9 to represent all values of x that satisfy the inequality.
The final solution indicates that any x between -5 and 9 (not including -5 and 9 themselves) satisfies the given absolute value inequality.