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.
From Two Separate Parts to One Chain
Solving a Three-Part Inequality Step by Step
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As you saw in the previous lesson, an "and" inequality like 3x+2>−4and4x−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:
1<x+3<7
This single line says exactly the same thing as "1<x+3andx+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:
−1<2x+3≤9
Step 1 — Remove the constant from the middle. Subtract 3 from all three parts:
−1−3<2x+3−3≤9−3−4<2x≤6
Step 2 — Remove the coefficient. Divide all three parts by 2:
2−4<22x≤26−2<x≤3
The solution tells us that x can be any value greater than−2 and less than or equal to3. Notice that the two inequality symbols (< and ≤) carried through unchanged because we divided by a positive number.
Here is a quick summary of the process:
Apply addition or subtraction to all three parts to remove the constant term beside the variable.
Apply multiplication or division to all three parts to make the variable's coefficient equal to 1.
Read the final chain to identify the solution's boundaries and whether each is included or excluded.
Dividing by a Negative: Flipping Both Signs
Applying Three-Part Inequalities to Real Situations
Conclusion and Next Steps
In this lesson, you learned that a three-part inequality is a compact way to write an "and" compound inequality, with the variable expression placed in the middle between two bounds. Solving it requires applying every operation to all three parts simultaneously so the chain stays balanced. When dividing or multiplying by a negative, remember to flip both inequality symbols and then rewrite the result with the smaller value on the left.
Now it is time to put these ideas into action! The upcoming practice exercises will guide you through completing partial solutions, solving three-part inequalities on your own, handling negative coefficients, and even setting up a real-world temperature problem from scratch. Let's jump in and build that confidence!
As you learned in earlier coursework, dividing (or multiplying) by a negative number reverses the direction of an inequality symbol. In a three-part inequality, this rule applies to both symbols at the same time. Let's see how this works.
−7≤−3x+2<11
Step 1 — Subtract2 from all three parts:
−9≤−3x<9
Step 2 — Divide all three parts by −3. Because we are dividing by a negative, we flip both inequality symbols:
−3−9≥x>−393≥x>−3
This reads "3 is greater than or equal to x, which is greater than −3." While mathematically correct, it is conventional to write the smaller value on the left. We simply reverse the entire chain:
−3<x≤3
Two important things happened in that final step:
Both inequality symbols flipped when we divided by the negative number.
We reordered the chain so the smaller boundary sits on the left, which is the standard way to present the answer.
Forgetting to flip both symbols is one of the most common mistakes in three-part inequalities, so it is worth pausing to double-check whenever a negative divisor or multiplier is involved.
Three-part inequalities show up naturally whenever a quantity must stay between two limits. Imagine a room starts at 60°F and rises by 1.5°F each hour while a heater runs. The temperature after h hours is 1.5h+60. If the room should be at least66°F but no more than75°F, we can express this as:
66≤1.5h+60≤75
Step 1 — Subtract60 from all three parts:
6≤1.5h≤15
Step 2 — Divide all three parts by 1.5 (positive, so no flipping):
4≤h≤10
The heater should run for at least 4 hours and at most 10 hours. Notice how naturally the problem fits the three-part format: one formula, two boundaries, one chain. Whenever you see a phrase like "must stay between," "should be at least … but no more than," or "ranges from … to," a three-part inequality is likely the right tool.
