Quadratic Inequalities

Quadratic Inequalities

A quadratic inequality asks where a parabola is above, below, or on the x-axis. Solve the related equation to find critical roots, sketch the parabola, then choose intervals that satisfy the sign. Include or exclude roots according to $\leq$ or $<$; a number-line or sign diagram makes the interval choice visible.

Worked reasoning

For $x^2-5x+6>0$, roots are 2 and 3. The parabola opens upward, so it is positive outside the roots: $x<2$ or $x>3$.

Exam method

1. Bring all terms to one side. 2. Factor or solve for roots. 3. Sketch/sign-test intervals and write the correct inequalities.

Sign-chart habit

Use one test value from every interval separated by the roots. For $(x-1)(x-4)$, try 0, 2 and 5; the signs reveal negative, positive/negative, positive products without relying on memory. Then revisit the inequality sign: strict inequalities exclude roots, while $\leq$ and $\geq$ include them. A final number-line sketch with open or closed circles communicates the answer precisely and prevents losing endpoint marks.

One-minute retrieval: Quadratic Inequalities

Close the worked solution. From memory, state its central rule, name one condition that makes it valid, and reconstruct one check that would catch a typical exam error.