Linear and Quadratic Inequalities: Methods and Examples

An inequality compares quantities without claiming that they are equal. Its solution is usually a set of values rather than one number. Strict inequalities exclude equality, while non-strict inequalities include it. Algebraic manipulation locates the boundary; a number line, interval notation, or sign chart then records which side of that boundary satisfies the original comparison.

Linear inequalities behave like equations until a sign reverses

Adding or subtracting the same quantity on both sides preserves the order. Multiplying or dividing by a positive number also preserves it. A negative factor reverses the order because it reverses positions on the number line. For example, 2 is less than 5, but multiplying both by negative one gives negative 2 greater than negative 5.

An open endpoint represents a strict comparison such as $x<7$. A closed endpoint represents $x\le 7$. Infinity is never an included endpoint, so interval notation always uses a parenthesis next to $\infty$ or $-\infty$. When two conditions must both hold, take their intersection. When either condition is allowed, take their union.

Worked linear example: isolate the variable and check the boundary

Solve $-3x+4>10$. Subtract 4 to get $-3x>6$. Dividing by negative 3 reverses the comparison, giving $x<-2$. The boundary value negative 2 makes both sides equal, so it must be excluded. Testing negative 3 gives 13 on the left, which is greater than 10 and confirms the chosen side.

A quadratic inequality is a question about signs

Move every term to one side and find the real zeros of the resulting quadratic. Those zeros divide the number line into intervals on which a continuous polynomial cannot change sign without passing through zero. Factorization, the quadratic formula or a graph can locate the zeros. Then test one value in each interval or use the direction of the parabola.

For $x^2-5x+6<0$, factor the polynomial as $(x-2)(x-3)$. The upward-opening parabola is below the horizontal axis only between its two roots, so the solution is the open interval from 2 to 3. If the comparison were less than or equal to zero, both roots would be included.

See how a quadratic changes sign

Move the sliders. The graph and interval summary update together.

Coefficient aCoefficient bCoefficient c

Denominators and square roots add domain boundaries

For a rational inequality, values that make a denominator zero are excluded even if a simplified expression appears to accept them. Zeros of the numerator and denominator both belong in the sign chart, but only numerator zeros may be included in a non-strict solution. For an even-indexed root, require the radicand to be nonnegative. Squaring both sides can introduce extraneous values, especially when one side may be negative, so substitute candidates back into the original statement.

A graph verifies the interval without replacing the algebra

The graph of $f\left(x\right)>0$ is above the horizontal axis on the solution intervals. This makes a graph a useful check, but screen resolution can hide a repeated root or a very narrow interval. Exact roots and domain restrictions should determine the final notation. Use the interactive view to build intuition, then retain the symbolic endpoints in the answer.