Inequalities: complete GMAT guide

You can add or subtract the same quantity from both sides without changing the direction. You can multiply or divide both sides by a positive quantity without changing the direction. Multiplying or dividing by a negative quantity reverses the inequality:

The same rule applies to $\le$ and $\ge$. If you swap the two sides of an inequality, reverse the sign: $3 \lt x$ is the same as $x \gt 3$.

A compound inequality such as $-2 \lt 3x+1 \le 10$ describes a range. Perform the same operation on all three parts:

$-3 \lt 3x \le 9$, so $-1 \lt x \le 3$.

An "and" inequality requires both conditions to hold, so the solution is the overlap. An "or" inequality requires at least one condition, so the solution is the union. For example, $x \lt -1$ or $x \gt 4$ cannot be written as a single interval; it is two separate rays.

The expression $|x-a|$ measures the distance from $x$ to $a$. This interpretation makes absolute value inequalities easier:

• $|x-a| \lt b$ means the distance is less than $b$, so $-b \lt x-a \lt b$, or $a-b \lt x \lt a+b$.
• $|x-a| \gt b$ means the distance is greater than $b$, so $x-a \lt -b$ or $x-a \gt b$.

The inequality $|x-3| \gt 5$ gives $x-3 \lt -5$ or $x-3 \gt 5$, so $x \lt -2$ or $x \gt 8$.

To solve a quadratic inequality, first find the roots, then test the intervals between them. For $x^2-5x+6 \lt 0$, factor as $(x-2)(x-3) \lt 0$. The roots are 2 and 3. The product is negative between the roots and positive outside them. The solution is $2 \lt x \lt 3$.

A sign chart keeps the intervals organized:

1. Mark the roots on a number line.
2. Test one value in each interval.
3. Select the intervals that satisfy the inequality.
4. Decide whether the roots themselves are included, based on $\lt$ or $\le$.

When an inequality contains a fraction, the denominator's sign matters. For $\frac{x-1}{x+2} \gt 0$, the fraction is positive when the numerator and denominator have the same sign. The critical points are $x=1$ and $x=-2$. Test the intervals $(-\infty,-2)$, $(-2,1)$, and $(1,\infty)$. The fraction is positive on $(-\infty,-2)$ and $(1,\infty)$. The point $x=-2$ is excluded because the denominator is zero.

A common shortcut is to multiply both sides by the square of the denominator, which is positive, but this usually creates a higher-degree inequality. A sign chart is more reliable.

Data Sufficiency inequality questions often ask a yes/no question, such as "Is $x \gt 0$?" A statement is sufficient only if it forces a single answer for every allowed value. If a statement gives $x^2 \gt 4$, then $x \gt 2$ or $x \lt -2$. The statement does not determine whether $x$ is positive, so it is not sufficient for that question.

Test cases with positive and negative values, fractions between 0 and 1, and zero when they are allowed. Many inequality statements fail because a fraction behaves differently from an integer.

On a number line, use an open circle for $\lt$ or $\gt$ and a closed circle for $\le$ or $\ge$. Shade the direction that satisfies the inequality. For two-variable inequalities, graph the boundary line and shade one side. A dashed boundary means the line is not included; a solid boundary means it is.

When an inequality looks complicated, test values from each interval. Choose a value less than the smaller root, a value between the roots, and a value greater than the larger root. Use simple numbers such as -10, 0, and 10, but avoid the roots themselves, where the expression equals zero.

For a quadratic with a negative leading coefficient, the sign pattern reverses. The parabola opens downward, so the expression is positive between the roots and negative outside them. Always check the leading coefficient before writing the intervals.

The expression $|x-a|$ is the distance from $x$ to $a$ on the number line. A statement such as $|x-5| \lt 2$ means that $x$ is within 2 units of 5, so $3 \lt x \lt 7$. A statement such as $|x-5| \gt 2$ means that $x$ is more than 2 units away from 5, so $x \lt 3$ or $x \gt 7$.

This geometric interpretation is often faster than algebra, especially when the inequality involves a distance from a variable point. Translate the statement into a distance sentence before writing the two cases.

1. Simplify each side without multiplying or dividing by a variable of unknown sign.
2. If you multiply or divide by a negative number, reverse the inequality.
3. For absolute value, rewrite as a distance statement or split into two cases.
4. For quadratics, find the roots and test intervals.
5. For fractions, identify excluded values and use a sign chart.
6. Check the endpoints and whether they are included.

• Forgetting to reverse the inequality when multiplying or dividing by a negative number.
• Treating "and" and "or" as the same condition.
• Including a value that makes a denominator zero.
• Assuming $x^2 \gt a$ means $x \gt \sqrt{a}$ without considering the negative root.
• Testing only integers when fractions are allowed.
• Writing a compound inequality in the wrong order.