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When dealing with absolute value inequalities, it's important to understand that absolute values measure distance from zero on the number line. So, the inequality \(|3-2x| \leq 5\) is asking: what values of \(x\) will keep the expression \(3-2x\) within 5 units of zero?

To solve absolute value inequalities, you split the inequality into two separate cases. This is because the expression inside the absolute value, \(3-2x\), can either be positive or negative. To address both cases, we write:

• \(3-2x \geq -5\)
• \(3-2x \leq 5\)

Each of these represents a scenario for how the expression can be within 5 units from zero, either on the positive or negative side of the number line. Solving these inequalities gives the complete solution for the variable \(x\).
Well done breaking these into inequalities allows for straightforward solution-finding, which then leads us to interval notation.

Interval notation is a way of expressing a set of values, typically solutions to inequalities, compactly and clearly. Using interval notation allows you to list ranges of numbers without having to write out complex inequality symbols repeatedly.

For instance, after solving the absolute value inequality, we find that \(-1 \leq x \leq 1\). In interval notation, this range is written as \([-1, 1]\).

Here's what the symbols mean:

• Brackets \([\text{or} ]\) indicate that the endpoint is included in the set, known as "inclusive" bounds.
• Parentheses \((\text{or} )\) indicate that the endpoint is not included, which are "exclusive" bounds.

By writing \([-1, 1]\), we communicate that \(x\) includes all real numbers between \(-1\) and \(1\), and \(-1\) and \(1\) themselves.

A solution set is a collection of values that satisfy a given equation or inequality. When you solve an inequality like \(|3-2x| \leq 5\), finding the solution set means finding all possible values of \(x\) that make the inequality true.

In our example, after solving both parts of the broken-down inequality, we discover that the solution set is the range of \(x\) values from \(-1\) to \(1\), inclusive. In concise mathematical terms, this is expressed in interval notation as \([-1, 1]\).

Understanding solution sets is crucial because:

• They provide the complete answer to the inequality problem.
• They help in visualizing which numbers satisfy an inequality or an equation on a number line.

By mastering how to find and express solution sets, you ensure clarity in solving and communicating about equations and inequalities.