Solving Three Part Inequalities

Welcome back to Solving Compound Inequalities! You are now on the third lesson of this course, and the skills you have been building are about to pay off in a big way.

In the previous lesson, we solved "and" compound inequalities by tackling each part separately and then finding the overlap. Many of those solutions turned out to be bounded regions, with the variable squeezed between a lower bound and an upper bound. In this lesson, we will learn a streamlined method for solving that same type of problem when it is written in a compact, three-part form. Instead of splitting the work into two separate inequalities, we will apply operations to all three parts at once, keeping everything aligned in a single chain.

As you saw in the previous lesson, an "and" inequality like $3x + 2 > -4$ and $4x - 1 < 15$ was solved by working on each piece individually and then combining the results into $-2 < x < 4$. That approach works perfectly, but many textbooks and real-world problems present the same idea in a more compact layout called a three-part (or chained) inequality.

A three-part inequality places the variable expression in the middle, flanked by a lower bound on the left and an upper bound on the right. For example:

This single line says exactly the same thing as "$1 < x + 3$ and $x + 3 < 7$." The chained form is simply a shorthand that keeps both conditions visible in one statement. Because both conditions share the same middle expression, we can solve them simultaneously rather than one at a time — and that is the key advantage.

Think of it like adjusting a thermostat range. If the acceptable temperature must stay between two limits, you would not think of the lower limit and the upper limit as two unrelated constraints. They naturally belong together, and the three-part form reflects that by keeping everything on one line.

The goal is to isolate the variable in the middle while keeping all three parts connected. Every operation we apply must hit the left part, the middle part, and the right part equally. This preserves the balance across the entire chain.

Let's walk through an example:

Step 1 — Remove the constant from the middle. Subtract $3$ from all three parts:

Step 2 — Remove the coefficient. Divide all three parts by $2$:

The solution tells us that $x$ can be any value greater than $-2$ and less than or equal to $3$. Notice that the two inequality symbols ($<$ and $\leq$) carried through unchanged because we divided by a positive number.

Here is a quick summary of the process:

1. Apply addition or subtraction to all three parts to remove the constant term beside the variable.
2. Apply multiplication or division to all three parts to make the variable's coefficient equal to $1$.
3. Read the final chain to identify the solution's boundaries and whether each is included or excluded.