Question

1. |6y - 2| + 4 < 22

Explanation:

Step1: Isolate the absolute value

Subtract 4 from both sides of the inequality \(|6y - 2|+4\lt22\).
We get \(|6y - 2|\lt22 - 4\), which simplifies to \(|6y - 2|\lt18\).

Step2: Solve the compound inequality

The absolute value inequality \(|a|\lt b\) (where \(b\gt0\)) is equivalent to \(-b\lt a\lt b\).
So for \(|6y - 2|\lt18\), we have \(- 18\lt6y - 2\lt18\).

Step3: Solve for y (left inequality)

Add 2 to all parts of the compound inequality. First, for the left - hand side: \(-18 + 2\lt6y-2 + 2\), which gives \(-16\lt6y\).
Then divide both sides by 6: \(\frac{-16}{6}\lt y\), simplifying the fraction \(\frac{-16}{6}\) to \(\frac{-8}{3}\), so \(-\frac{8}{3}\lt y\).

Step4: Solve for y (right inequality)

For the right - hand side of the compound inequality \(6y-2 + 2\lt18 + 2\), we get \(6y\lt20\).
Divide both sides by 6: \(y\lt\frac{20}{6}\), simplifying the fraction \(\frac{20}{6}\) to \(\frac{10}{3}\), so \(y\lt\frac{10}{3}\).

Answer:

The solution to the inequality \(|6y - 2|+4\lt22\) is \(-\frac{8}{3}\lt y\lt\frac{10}{3}\)