Special Inequality Solutions

You have made it to the final lesson of Solving Multi-Step Inequalities — congratulations on getting this far! Over the previous four lessons, we built a powerful toolkit: two-step solving, combining like terms, expanding with the distributive property, and collecting variable terms from both sides. Every problem we have solved ended with a clean result like $x < 4$ or $x \geq 3$, where one boundary value neatly divides the number line into solutions and non-solutions. In this lesson, we meet the two surprising cases where that does not happen. Sometimes an inequality has no solution at all, and sometimes every real number is a solution. Let's find out how to recognize and interpret both.

In the last lesson, we practiced moving variable terms to one side of the inequality. Usually that left us with a nonzero variable term we could isolate. But what happens when the variable terms on both sides are identical? When we subtract one from the other, they cancel completely, and we are left with a statement that contains only numbers — no $x$ in sight.

For example, simplifying might lead to something like $5 > 2$ or $3 < 0$. There is no variable left to solve for, so the statement is either true or false on its own, regardless of what $x$ might be. These are the two special outcomes we will study in this lesson: always true and always false.

Let's work through an example step by step. Consider:

Step 1 — Subtract $2x$ from both sides to collect the variable terms:

The variable is gone. We are left with $9 < 1$, which is simply false. Nine is never less than one, no matter what value $x$ takes.

Because the original inequality simplifies to a false statement, there is no value of $x$ that can make it true. We say the solution set is empty, or equivalently, there is no solution. A quick check confirms this: try $x = 0$ and we get $9 < 1$ (false); try $x = 100$ and we get $209 < 201$ (still false). No value works.

Now let's see the opposite outcome. Consider:

Step 1 — Distribute the $3$ on the left side:

Step 2 — Subtract $3x$ from both sides:

Again, the variable has disappeared. This time the remaining statement, $-6 \leq 5$, is true. Negative six is always less than or equal to five.

Because the original inequality simplifies to a true statement, every value of $x$ makes it true. The solution set is all real numbers. Plug in $x = 0$ and you get $-6 \leq 5$ (true); plug in $x = -1000$ and you get $-3006 \leq -2995$ (true). Any value we choose will work.