Absolute Value Inequalities and Neighborhoods

For \(a > 0\), the inequality \(|x| < a\) is equivalent to:

For \(a > 0\), the inequality \(|x| > a\) is equivalent to:

Find the values of x satisfying \(\left|\dfrac{3x+1}{2}\right| < 1\). Give the upper bound of the resulting interval, as a fraction.

Find the values of x satisfying \(\left|\dfrac{3x+1}{2}\right| < 1\). Give the upper bound of the resulting interval, as a fraction.

Find the values of x satisfying \(\left|\dfrac{3x+1}{2}\right| < 1\). Give the lower bound of the resulting interval.

Find the values of x satisfying \(\left|\dfrac{3x+1}{2}\right| < 1\). Give the lower bound of the resulting interval.

The inequality \(|x - a| < c\) describes an interval centered at a with endpoints:

The \(\varepsilon\)-neighborhood of a point a is defined as:

The deleted neighborhood of a is defined by which inequality?

In a deleted neighborhood of a, the point ______ itself is excluded from the set.

In a deleted neighborhood of a, the point ______ itself is excluded from the set.

Show that \(|(3x + 2) - 8| < 1\) is equivalent to \(|x - 2| < k\). What is the value of k?