Quadratic Inequalities Math 2 Unit 1

What is a quadratic inequality? A quadratic inequality looks just like a quadratic equation, but will have a less than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥) sign instead of the normal equals sign (=).

What kind of solutions do you get from quadratic inequalities? With quadratic inequalities, we look for solution sets (i.e. an interval for which the inequality is true) Examples of solution sets would be: X< 5 (every number less than 5 makes the inequality true) 3<x<25 (every number greater than 3 AND less than 25….so every number between 3 and 25)

Two Ways that We Will Write Solution Sets Using <,>,≤, ≥ signs like on the last slide. For example, 3 < x ≤ 8 Using interval notation. For example, (3,8]

Interval Notation Review Important Points to Remember: If the inequality had a ≥ or ≤, you will use brackets in your solution [ ] If the inequality had > or <, you will use parentheses ( ) Infinity always needs parentheses! For x < 5, we would write (-∞, 5) For x > -2, we would write (-2, ∞) For -7 < x < 3, we would write (-7, 3) For x ≥ 8, we would write [8, ∞)

Interval Notation Example 1 Write in interval notation: 4 < x < 13

Interval Notation Example 2

Interval Notation Example 3

Interval Notation Example 4 Write as an inequality, then graph on a number line. [ 11, 28 ]

Interval Notation Example 5 Write as an inequality, then graph on a number line. ( -∞, 15)

Interval Notation Example 6 Write as an inequality then graph on a number line. [ -5, ∞)

Solving Quadratic Inequalities 1.) Rewrite inequality as an equation. 2.) Set equation equal to zero. 3.) Solve equation by factoring or quadratic formula. These two solutions will be your CRITICAL POINTS. 4.) Draw a number line and plot your critical points. 5.) Identify the three intervals created by your critical points.

Solving Quadratic Inequalities cont. 6.) Pick a number from inside your first interval. DO NOT pick a critical point. Test this point by plugging the number in for x in the original inequality. If the number makes the inequality true, then that interval will be part of your solution set. To save yourself work, pick simple numbers to work with. (0, ±1, ±10, etc.) 7.) Do the same for the other two intervals. 8.) Write your solution set in interval notation.

Example 1 x2 + 5x + 6 ≥ 0 1.) Rewrite inequality as an equation. 2.) Set equation equal to zero. 3.) Solve equation by factoring or quadratic formula. These two solutions will be your CRITICAL POINTS. 4.) Draw a number line and plot your critical points. 5.) Identify the three intervals created by your critical points. 6.) Pick a number from inside your first interval. DO NOT pick a critical point. Test this point by plugging the number in for x in the original inequality. If the number makes the inequality true, then that interval will be part of your solution set. To save yourself work, pick simple numbers to work with. (0, ±1, ±10, etc.) 7.) Do the same for the other two intervals. 8.) Write your solution set in interval notation.

Example 2 2x2 – 11x ≤ -14 1.) Rewrite inequality as an equation. 2.) Set equation equal to zero. 3.) Solve equation by factoring or quadratic formula. These two solutions will be your CRITICAL POINTS. 4.) Draw a number line and plot your critical points. 5.) Identify the three intervals created by your critical points. 6.) Pick a number from inside your first interval. DO NOT pick a critical point. Test this point by plugging the number in for x in the original inequality. If the number makes the inequality true, then that interval will be part of your solution set. To save yourself work, pick simple numbers to work with. (0, ±1, ±10, etc.) 7.) Do the same for the other two intervals. 8.) Write your solution set in interval notation.

Example 3 3x2 – 8x > -4 1.) Rewrite inequality as an equation. 2.) Set equation equal to zero. 3.) Solve equation by factoring or quadratic formula. These two solutions will be your CRITICAL POINTS. 4.) Draw a number line and plot your critical points. 5.) Identify the three intervals created by your critical points. 6.) Pick a number from inside your first interval. DO NOT pick a critical point. Test this point by plugging the number in for x in the original inequality. If the number makes the inequality true, then that interval will be part of your solution set. To save yourself work, pick simple numbers to work with. (0, ±1, ±10, etc.) 7.) Do the same for the other two intervals. 8.) Write your solution set in interval notation.

Example 4 6x2 > 11x - 3 1.) Rewrite inequality as an equation. 2.) Set equation equal to zero. 3.) Solve equation by factoring or quadratic formula. These two solutions will be your CRITICAL POINTS. 4.) Draw a number line and plot your critical points. 5.) Identify the three intervals created by your critical points. 6.) Pick a number from inside your first interval. DO NOT pick a critical point. Test this point by plugging the number in for x in the original inequality. If the number makes the inequality true, then that interval will be part of your solution set. To save yourself work, pick simple numbers to work with. (0, ±1, ±10, etc.) 7.) Do the same for the other two intervals. 8.) Write your solution set in interval notation.

Complete Assignment #1 Pg. 101: 22-31 Be prepared to share your answers. Be prepared to turn in your paper before you leave.