Solve the inequality. Graph the solution.

Question

StartAbsoluteValue x minus 2 EndAbsoluteValuex−2less than<77
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Answer 1

To solve the inequality \( |x - 2| < 7 \), we need to break it down into two parts because the absolute value can be rewritten as two inequalities.

The expression \( |x - 2| < 7 \) means that the distance between \( x \) and 2 is less than 7. This gives us the following compound inequality:

\[ -7 < x - 2 < 7 \]

Step 1: Solve the compound inequality

1. Add 2 to all parts of the inequality: \[ -7 + 2 < x < 7 + 2 \]

   This simplifies to: \[ -5 < x < 9 \]

Step 2: Write the solution in interval notation

The solution to the inequality can be written in interval notation as: \[ (-5, 9) \]

Step 3: Graph the solution

To graph the solution \( (-5, 9) \):

1. Draw a number line.
2. Mark the points -5 and 9.
3. Use open circles at -5 and 9 to indicate that these endpoints are not included in the solution.
4. Shade the line between -5 and 9 to indicate all the values \( x \) can take.

The graph would look like this:

<----(-----|-----)---->
    -5     0     9

Conclusion

The solution to the inequality \( |x - 2| < 7 \) is the interval \( (-5, 9) \), and it is represented graphically on a number line.