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Questions and Answers
What is the form of a quadratic inequality?
What is the form of a quadratic inequality?
How can the solutions to a quadratic inequality be found?
How can the solutions to a quadratic inequality be found?
What is the first step in solving an inequality?
What is the first step in solving an inequality?
What should be done when multiplying or dividing both sides of an inequality by a negative number?
What should be done when multiplying or dividing both sides of an inequality by a negative number?
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What is the purpose of the last step in solving an inequality?
What is the purpose of the last step in solving an inequality?
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What is the goal of solving an inequality?
What is the goal of solving an inequality?
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Study Notes
Quadratic Inequalities
- A quadratic inequality is an inequality of the form ax^2 + bx + c > 0, ax^2 + bx + c ≥ 0, ax^2 + bx + c < 0, or ax^2 + bx + c ≤ 0, where a, b, and c are real numbers and a ≠ 0.
- The solutions to a quadratic inequality can be found by:
- Factoring the left-hand side, if possible
- Using the Quadratic Formula to find the roots of the related quadratic equation
- Graphing the related quadratic function on a number line
Solving Inequalities
- Solving an inequality involves finding the values of the variable that make the inequality true.
- The steps to solve an inequality are:
- Simplify the inequality by combining like terms and eliminating any parentheses or other grouping symbols.
- Add or subtract the same value to both sides of the inequality to isolate the variable term.
- Multiply or divide both sides of the inequality by a coefficient of the variable, if necessary, to solve for the variable.
- Note: When multiplying or dividing both sides by a negative number, the direction of the inequality symbol must be flipped.
- Check the solution by plugging it back into the original inequality.
Properties of Inequalities
- The following properties can be used to solve inequalities:
- Addition Property: If a > b, then a + c > b + c
- Subtraction Property: If a > b, then a - c > b - c
- Multiplication Property: If a > b and c > 0, then ac > bc
- If a > b and c < 0, then ac < bc
- Transitive Property: If a > b and b > c, then a > c
Graphing Inequalities
- Inequalities can be graphed on a number line to visualize the solution.
- Open circles or parentheses are used to indicate strict inequalities (e.g., < or >), while closed circles or brackets are used to indicate inclusive inequalities (e.g., ≤ or ≥).
- The direction of the inequality symbol indicates the direction of the solution on the number line.
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Description
Learn about quadratic inequalities and how to solve them using factoring, quadratic formula, and graphing. Practice solving inequalities with this quiz.
