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10/26/2015 Warmup: Solve and graph ο 5π₯ β 6 < 4 ο β5 β€ 3 4 β π₯ ο 3π₯ β 2 β€ 4π₯ + 9 ο 7 π₯ β 11 β€ 3π₯ + 4π₯ β 3 Section 2.5 Compound Inequalities 1 10/26/2015 What is a combined compound inequality? ο ο ο ο ο ο ο 1) You take two inequalities, and combine them into one. Like compound sentences. Example. 5>π₯ πππ π₯ > β5 We can write this as a compound inequality. Make sure that the signs face β β5 < π₯ < 5 The graph is of the two original inequalities on the same number line. Example on next slide. β5 < π₯ < 5 We graph it like it looks. π₯ is in between those two numbers! The same βfilled in circleβ and βhollow circleβ process applies! π₯ > β5 You Graph 5>π₯ β6 β€ π₯ β€ 10 2 10/26/2015 Important: ο ο ο ο ο ο ο Compound Inequalities that are combined always have the symbols facing the same way, and generally facing right. The variables will always be in the middle. They go from small to large. Examples. β21 β€ π₯ < 23 .5 < π₯ β€ 1 β4 β€ π₯ β€ 80 β2 β€ π₯ β 6 < 8 ο Are these written correctly? 1) 4 β€ π < β1 ο 2) π < 45 < 99 ο 3) 19 > π₯ < 11 ο 4) 2 > π₯ > β3 ο Oh heck no! 3 10/26/2015 Uncombined Compound inequalities ο ο ο ο Compound Inequalities that are not combined have an βorβ between two inequalities. These should always go in opposite directions on the graph. Examples. x > 5 ππ π₯ < β5 x > 4 ππ π₯ β€ β5 Graphing Uncombined Compound inequalities Examples 5 < π₯ ππ π₯ β€ β5 π₯ β€ β5 You Graph 6β€π₯ 5<π₯ ππ 2π₯ β€ β10 4 10/26/2015 Interpreting a graph. We need to make sure we are able to interpret a graph and find itβs inequality. Example. Solving compound Inequalities The formal way. Example 13 < 4 + 3π₯ β€ 55 STEP 1) write as 2 inequalities 13 < 4 + 3π₯ πππ 4 + 3π₯ β€ 55 STEP 2) Solve Each separately 13 < 4 + 3π₯ πππ 4 + 3π₯ β€ 55 9 < 3π₯ πππ 3π₯ β€ 51 3 < π₯ πππ π₯ β€ 17 ο STEP 3) COMBINE AGAIN 3 < π₯ β€ 17 You did it! Now that weβve got that, Let me show you a faster way! 5 10/26/2015 The Fast Way of solvingβ¦ To solve the compound inequality, we have to do inverse properties to BOTH sides. ο 4 < π₯ β 3 < 13 +3 +3 +3 Now we add to BOTH sides! ο 7 < π₯ < 16 We are done! ο Watch out for flipping the sign. That still applies! ο ο ο ο ο ο Example 2) Find the solutions and graph. β5 β€ 7 β 2π₯ < 11 β7 β7 β7 Sub on all sides. β12 β€ β2π₯ < 4 Divide all sides by -2. FLIPS THAT SIGN! 6 β₯ π₯ > β2 Then write it the correct way β2 < π₯ β€ 6 One more Example with sign flippage. 6 10/26/2015 Try this for yourself! #2 is beastly ο 1) 22 β€ 4 β 6π < 64 ο 2) 45 < 13 + 15π₯ < 135 ο P.101 AP 2.6 All P.102 PS 2.6 #1-4 ο ο ο Check Yoself 1) β10 < π β€ β3 2 2 2) 2 15 < π₯ <815 7