Question

homework 1.7
question 1 of 10 (1 point) | question attempt: 1 of unlimited
(a) |x| = 3
(b) |x| > 3
(c) |x| < 3
part 1 of 5
(a) the solution set for |x| = 3 is {-3, 3}.
part 2 of 5
(b) the solution set for |x| > 3 is (-∞, -3) ∪ (3, ∞).
part: 2 / 5
part 3 of 5
graph the solution set for |x| > 3.

Explanation:

Step1: Understand the inequality

The inequality \(|x| > 3\) means that the distance of \(x\) from 0 on the number line is greater than 3.

Step2: Identify the regions

For a number \(x\), if \(|x| > 3\), then either \(x > 3\) (because numbers to the right of 3 on the number line are more than 3 units away from 0) or \(x < - 3\) (because numbers to the left of - 3 on the number line are more than 3 units away from 0).

Step3: Graphing the solution

• For the region \(x > 3\): We draw an open circle at \(x = 3\) (since \(x = 3\) does not satisfy \(|x|>3\)) and draw an arrow to the right of 3 to represent all numbers greater than 3.
• For the region \(x < - 3\): We draw an open circle at \(x=-3\) (since \(x = - 3\) does not satisfy \(|x|>3\)) and draw an arrow to the left of - 3 to represent all numbers less than - 3.

Answer:

To graph \(|x|>3\):

1. On the number line, place an open circle at \(x = 3\) (because \(x = 3\) does not satisfy \(|x|>3\)) and draw a ray (arrow) starting from the open circle at 3 and extending to the right (towards \(+\infty\)).
2. On the number line, place an open circle at \(x=-3\) (because \(x = - 3\) does not satisfy \(|x|>3\)) and draw a ray (arrow) starting from the open circle at - 3 and extending to the left (towards \(-\infty\)).