How do you solve and graph the absolute value inequality |2x – 4| < 14?

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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:

  1. For the left part:
    • -14 < 2x - 4
    • Add 4 to both sides:
    • -10 < 2x
    • Now divide both sides by 2:
    • -5 < x
  2. 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.

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