Absolute Value Inequality and Interval Notation
The given inequality is \( |x| \leq \frac{11}{3} \), and we are asked to express this set in interval notation and graph it on a number line.
First, let's solve the absolute value inequality:
An absolute value inequality like \( |x| \leq \frac{11}{3} \) means that the distance between \( x \) and 0 on the number line is less than or equal to \( \frac{11}{3} \). We break this into two separate inequalities:
1. \( x \leq \frac{11}{3} \), because \( x \) can be to the right of 0 and still be within \( \frac{11}{3} \) units of 0.
2. \( x \geq -\frac{11}{3} \), because \( x \) can also be to the left of 0 and be within \( \frac{11}{3} \) units of 0.
The solution to the inequality is the set of all \( x \) that satisfy both conditions, meaning \( x \) is between \( -\frac{11}{3} \) and \( \frac{11}{3} \), inclusive of both endpoints.
The interval notation for this inequality is \([- \frac{11}{3}, \frac{11}{3}] \).
In terms of graphing this set on a number line, we would mark and shade the interval from \( -\frac{11}{3} \) to \( \frac{11}{3} \), including the endpoints, which are marked with a closed dot to indicate that they are part of the solution set.
The grouping symbols for this interval notation would be ([]) which indicates that the set includes its endpoints.
The endpoints entered into the boxes as shown in the image should be:
- Left endpoint: \( -\frac{11}{3} \)
- Right endpoint: \( \frac{11}{3} \)
For the number line provided in part (b), the shaded region would start at \( -\frac{11}{3} \) and end at \( \frac{11}{3} \). You would ensure that there is a closed dot or a solid circle at each of the endpoints \( -\frac{11}{3} \) and \( \frac{11}{3} \) to depict that these points are included in the interval.
