9.8 Solve Quadratic Inequalities

Quadratic inequalities are like puzzles that ask which x-values make a parabola higher or lower than zero. You can solve them by graphing or using algebra. The key is finding where the parabola crosses the x-axis.

The discriminant helps predict what the solution will look like. It tells you if the parabola crosses the x-axis twice, once, or not at all. This info is super useful for figuring out which x-values satisfy the inequality.

Solving Quadratic Inequalities

Graphical solutions for quadratic inequalities

• Quadratic inequalities have the form $ax^2 + bx + c > 0$, $ax^2 + bx + c < 0$, $ax^2 + bx + c \geq 0$, or $ax^2 + bx + c \leq 0$ where $a$, $b$, and $c$ are real numbers, with $a \neq 0$ ($a$ cannot equal zero)
• Graph the corresponding quadratic function $y = ax^2 + bx + c$ to solve graphically
  • Find the x-intercepts (roots) of the quadratic function by setting $y = 0$ and solving for $x$
  • Determine the concavity of the parabola based on the sign of $a$ (upward if $a > 0$, downward if $a < 0$)
• Shade the region above the x-axis for inequalities with $>$, or below the x-axis for inequalities with $<$
  • The solution set includes all x-values in the shaded region ($x$-coordinates of points in the shaded area)
• Include the points on the parabola itself in the solution set for inequalities with $\geq$ or $\leq$, following the same shading rules as above
• Express the solution using interval notation to represent the range of x-values that satisfy the inequality

Algebraic techniques for quadratic inequalities

• Set the quadratic expression equal to zero and solve for the critical points (x-intercepts or roots) to solve algebraically
  • Apply factoring, the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, or other suitable methods to find the critical points
• Identify the concavity of the parabola based on the sign of $a$ (upward if $a > 0$, downward if $a < 0$)
• Construct a sign chart or number line, marking the critical points
  • Evaluate the quadratic expression at a test point in each interval to determine the sign (positive or negative) within that interval
• The solution set consists of the intervals where the sign aligns with the inequality sign for inequalities with $>$ or $<$
• Include the endpoints (critical points) in the solution set if the sign at the endpoint matches the inequality sign for inequalities with $\geq$ or $\leq$

Discriminant in quadratic inequality solutions

• The discriminant is the expression $b^2 - 4ac$, found under the square root in the quadratic formula
• The value of the discriminant determines the nature of the solutions to the corresponding quadratic equation $ax^2 + bx + c = 0$
  1. If $b^2 - 4ac > 0$ (positive discriminant), the quadratic equation has two distinct real solutions
  2. If $b^2 - 4ac = 0$ (zero discriminant), the quadratic equation has one repeated real solution
  3. If $b^2 - 4ac < 0$ (negative discriminant), the quadratic equation has no real solutions (only complex solutions)
• The discriminant can help predict the nature of the solution set for quadratic inequalities
  • A positive discriminant typically results in a solution set consisting of two distinct intervals (or one interval and all real numbers, depending on the inequality sign and concavity)
  • A zero discriminant usually leads to a solution set consisting of a single interval (or all real numbers except one point, depending on the inequality sign and concavity)
  • A negative discriminant implies the solution set will either be all real numbers or the empty set, depending on the inequality sign and concavity

Related Inequality Types

• Compound inequalities: Involve multiple inequality statements combined with "and" or "or" operators, which may include quadratic expressions
• Absolute value inequalities: Can be rewritten as two separate inequalities, potentially involving quadratic expressions
• Polynomial inequalities: A broader category that includes quadratic inequalities as well as higher-degree polynomial expressions