How can I solve and graph the absolute value inequality |2x – 4| < 8?

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To solve the absolute value inequality |2x – 4| < 8, we need to break it down into two separate inequalities. The absolute value expression |A| < B indicates that the value of A lies between -B and B. In this case, our A is 2x – 4 and our B is 8.

We can set up the two inequalities as follows:

  1. 2x – 4 < 8
  2. 2x – 4 > -8

Now, let’s solve each inequality separately.

First Inequality: 2x – 4 < 8

1. Add 4 to both sides:

2x < 12

2. Divide both sides by 2:

x < 6

Second Inequality: 2x – 4 > -8

1. Add 4 to both sides:

2x > -4

2. Divide both sides by 2:

x > -2

Now, we combine the results of both inequalities:

-2 < x < 6

Interval Notation:

The solution can be written in interval notation as:

( -2, 6 )

Graphing the Solution:

To graph this inequality on a number line:

  • Draw a number line.
  • Place open circles at -2 and 6 (indicating these points are not included in the solution).
  • Shade the region between -2 and 6.

This shaded region represents all the values of x that satisfy the inequality |2x – 4| < 8.

Conclusion:

The solution to the absolute value inequality |2x – 4| < 8 is -2 < x < 6, and it can be represented graphically by a number line with open circles on -2 and 6 and shading between these points.

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