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Question

Part 1
Solve the inequality. Graph the solution.
StartAbsoluteValue x plus 2 EndAbsoluteValue greater than or equals 7x+2≥7
Question content area bottom
Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer​ box(es) to complete your choice.

Answer 1

To solve the inequality $|x+2|\ge 7$, we will break it down into two cases based on the properties of absolute values.

Case 1: $x+2\ge 7$

1. Solve the inequality: $$x+2\ge 7$$ $$x\ge 7-2$$ $$x\ge 5$$

Case 2: $x+2\le -7$

1. Solve the inequality: $$x+2\le -7$$ $$x\le -7-2$$ $$x\le -9$$

Combine the Solutions

From the two cases, we have:

1. $x\ge 5$
2. $x\le -9$

Solution in Interval Notation

The complete solution set can be written in interval notation as: $$(-\mathrm{\infty },-9]\cup [5,\mathrm{\infty })$$

Graphing the Solution

To graph the solution:

• Draw a number line.
• Shade the region to the left of -9 (including -9, which is a closed circle).
• Shade the region to the right of 5 (including 5, which is also a closed circle).

Select the Correct Choice

The final answer is: $$x\le -9\text{or}x\ge 5$$ Express your answer in interval notation: $$(-\mathrm{\infty },-9]\cup [5,\mathrm{\infty })$$