Solving inequalities: how to solve and graph them

An inequality compares two expressions that may not be equal, using $<$ (less than), $>$ (greater than), $\le$ (less than or equal to) or $\ge$ (greater than or equal to). Where an equation like $x + 4 = 9$ has one answer, an inequality like $x + 4 < 9$ has a whole range of answers: every number less than 5.

The good news: if you can solve equations, you already know how to do inequalities. Solving inequalities works almost exactly like solving equations — add, subtract, multiply or divide both sides by the same amount — with one important exception.

The one rule that is different

When you multiply or divide both sides by a negative number, reverse the inequality sign. Why? $2 < 3$ is true, but multiplying both sides by $-1$ gives $-2$ and $-3$, and $-2 > -3$. Multiplying by a negative flips the order of numbers, so the sign must flip too.

How to solve an inequality, step by step

1. Simplify each side (expand brackets, collect like terms).
2. Get the variable terms on one side and the numbers on the other.
3. Divide by the coefficient of the variable — and reverse the sign if it is negative.
4. Write the answer, graph it, and test one number from the answer in the original inequality.

The variable on both sides

Collect the variable on whichever side keeps its coefficient positive — then you never have to flip the sign.

Inequalities in words

• “at least”, “no less than”, “minimum” → $\ge$
• “at most”, “no more than”, “maximum” → $\le$
• “more than”, “over”, “exceeds” → $>$
• “less than”, “under”, “fewer than” → $<$

How to graph an inequality on a number line

• Put a dot on the boundary number.
• Use an open (hollow) dot for $<$ or $>$, because the number itself is not included.
• Use a closed (filled) dot for $\le$ or $\ge$, because it is included.
• Shade the side containing the solutions: left for “less than”, right for “greater than”.

For $x \le -4$: a filled dot at $-4$, shaded to the left. For $-2 \le x < 3$: a filled dot at $-2$, an open dot at 3, and shading between them. In interval notation these are $(-\infty, -4]$ and $[-2, 3)$ — square brackets for included ends, round ones for excluded ends and for infinity.

How to graph linear inequalities in two variables

An inequality like $y > 2x + 1$ describes a region of the coordinate plane, not a single line. To graph the linear inequality:

1. Draw the boundary line $y = 2x + 1$: dashed for $<$ or $>$, solid for $\le$ or $\ge$.
2. Pick a test point not on the line — $(0, 0)$ is easiest. Here $0 > 2(0) + 1$ is false.
3. Shade the side that does not contain the test point when the test is false (above the line here), or the side that does when it is true.

Common mistakes

• Forgetting to reverse the sign after multiplying or dividing by a negative.
• Using the wrong dot: open for $<$ and $>$, filled for $\le$ and $\ge$.
• Shading the wrong side. Always test one point.
• Treating “greater than” absolute value inequalities as one interval: $|x - 4| > 3$ means $x < 1$ or $x > 7$ — two pieces.