MISSION · Algebra
Solving and Graphing Inequalities
When the answer is a whole stretch of the number line
LEVEL 1 · EXPLORE
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LEVEL 2 · LEARN
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The idea in one sentence
An inequality is solved exactly like an equation, with one extra rule — and its answer is a range rather than a single number.
An answer that’s a region
\(x + 3 = 7\) has one answer: 4. \(x + 3 < 7\) has infinitely many — 3, 0, \(-12\), 3.999, and everything else below 4.
So the answer is written as a description of a set rather than a number: \(x < 4\). On a number line it’s a shaded ray, and the circle at the end tells you whether the boundary itself is included:
- Open circle for \(<\) or \(>\) — the boundary is not part of the answer.
- Filled circle for \(\le\) or \(\ge\) — it is.
That circle carries real information. “You must be under 18” and “you must be 18 or under” are different rules, and the only thing distinguishing them on the page is whether the dot is filled in.
Solve it like an equation
Every move you’d make on an equation works here: add to both sides, subtract from both sides, multiply or divide both sides.
\[3x + 5 \ge 20 \quad \Rightarrow \quad 3x \ge 15 \quad \Rightarrow \quad x \ge 5\]Same steps, same order, same reasoning. Undo the addition, then undo the multiplication.
The one extra rule
Multiplying or dividing by a negative flips the inequality sign.
Start with something obviously true: \(3 < 5\). Multiply both sides by \(-1\) and you’d get \(-3 < -5\), which is false — \(-3\) is the larger of the two. Flip the sign and it comes right: \(-3 > -5\).
Negating reflects the number line, and reflection reverses order. So:
\[-2x < 8 \quad \Rightarrow \quad x > -4\]Adding or subtracting a negative changes nothing about the sign. It’s only multiplying and dividing by one that flips it.
Reading them aloud
\(<\) is “less than”, \(>\) is “greater than”, \(\le\) is “less than or equal to”. The open end always faces the larger side, which is the quickest way to check you’ve written the one you meant.
Common mix-ups
- Forgetting to flip after dividing by a negative. The most common inequality error there is.
- Flipping after adding a negative, which is flipping when you shouldn’t.
- Using the wrong circle on the number line, which silently changes the answer.
- Giving a single number as the answer. \(x = 5\) answers an equation. An inequality’s answer is a range.
Check yourself
Solve \(-3x + 2 \le 11\) and describe the graph.
Subtract 2: \(-3x \le 9\). Divide by \(-3\) and flip: \(x \ge -3\). The graph is a filled circle at \(-3\) with shading to the right.
LEVEL 3 · REFLECT
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