Linear inequalities

A linear inequality looks like a linear equation, except that instead of an equals sign it carries one of $<$, $>$, $\le$, $\ge$:

The difference in the answer is fundamental. A linear equation usually has one solution; an inequality has a whole set of them. The answer is not a number but a condition such as $x > 3$.

What you may do to an inequality

Inequalities are rearranged just like equations, with one exception covered below. The following keep the inequality sign unchanged:

So terms may be moved to the other side (changing sign, as in equations), and both sides may be divided by a positive number.

The one exception: multiplying and dividing by a negative number

When both sides are multiplied or divided by a negative number, the inequality sign must be flipped:

Why? Because multiplying by a negative number reflects the number line about zero and reverses the order of numbers. $2 < 5$ is true, but after multiplying both sides by $-1$ we have $-2$ and $-5$, and $-2 > -5$. If the sign stayed as it was, a true statement would turn false.

Note that the flip is caused by multiplying or dividing by a negative number, not by a minus appearing somewhere in the problem. Moving $-3$ to the other side changes nothing about the sign.

The solution set on a number line

The solution of an inequality is easiest to see on a number line: mark the boundary number and take the whole ray on the side where the solutions lie.

Whether the boundary belongs to the set is decided by the kind of inequality:

• strict ($<$, $>$) — the boundary is not included; on a hand drawing it gets an open circle,
• non-strict ($\le$, $\ge$) — the boundary is included; a filled circle.

The same set is written more briefly as an interval:

A round bracket marks an end not in the set, a square bracket one that is. Infinity always takes a round bracket, because infinity is not a number.

Inequalities in practice

Inequalities answer questions of the "at most" and "at least" kind. A delivery costs 40 plus 6 per parcel, and the budget is 100. How many parcels fit the budget?

The solution set is every $x \le 10$, but a number of parcels is a non-negative whole number — so the answer is: at most ten. That is typical of word problems: mathematics gives the solution set, and the meaning of the problem narrows it down.

In the exercises below, enter the whole solution set, for example x > 3 or x ≥ 3 (x >= 3 works too). The boundary number alone is not enough — the direction counts as well.

Common mistakes

• Not flipping the sign after dividing by a negative number — the single most common mistake here.
• Flipping the sign when moving a term across — moving is addition, so the sign stays.
• Mixing up strict and non-strict — $x > 3$ excludes 3, $x \ge 3$ includes it.
• Answering with the boundary alone — $3$ is not the answer; $x > 3$ is.
• Ignoring what the problem is about — a number of parcels or people is never a fraction or negative.