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Category 5 - Algebra - Meet #3 – Study Guide Absolute Value; inequalities in one variable including interpreting line graphs Inequalities : Solving inequalities is exactly the same as solving equations with ONE EXCEPTION!!!!!! If at anytime in solving an inequality you need to multiply or divide both sides of the inequality by a negative number, you must reverse the < or >. ex. Solve x+5 ≤ −4 −2 ×–2 ×–2 5 – 2x > 13 –5 –5 – 2x > 8 ÷–2 ÷–2 x+5≥8 - 2 -2 x≥6 x<–4 Graphing Inequalities : x < 3 has a graph like The empty circle shows x ≠ 3 0 3 x ≥ – 4 has a graph like Filled in circle means x could = -4 –4 0 Absolute Value takes a value and tells you its distance from zero, or takes a DIFFERENCE and tells you the distance between the values you were taking the difference of. ex. − 4 since – 4 is 4 away from zero, the absolute value of – 4 is 4 ex. 5 since 5 is 5 away from zero, the absolute value of 5 is 5 ex. 0 since zero is zero away from zero the absolute value of zero is 0 ex. x − 3 = 6 is asking what numbers(x) are 6 away from 3. So 3 + 6 = 9 & 3 – 6 = – 3 You can be more formal about this by writing 2 separate equations : x – 3 = 6 or x – 3 = – 6 when you add 3 to both sides of each equation you still get x = 9 or – 3 ex. 2 x − 5 = 11 write two equations : 2x – 5 = 11 +5 +5 2x = 16 or ÷2 ÷2 x=8 or or 2x – 5 = – 11 +5 +5 2x = – 6 ÷2 ÷2 x= –3 When faced with an absolute value inequality it is most important that you do two things. The first is to think about what the inequality MEANS. The second is to check your solution to see if it makes sense. Using the absolute value equation above as an example and making it an inequality keeps the solution similar in come ways and different in others. First, what does 2 x − 5 < 11 mean? It means the value of 2x – 5 must have an absolute value less than 11. Well, what numbers have an absolute value less than 11? Numbers less than 11 do, but only up until they become less than -11. For example. – 12 has an absolute value of 12 which is not less than 11. So, what can 2x – 5 be? 2x – 5 < 11 AND 2x – 5 > – 11 (the AND means x must fulfill both inequalities) +5 +5 +5 +5 2x < 16 AND 2x > – 6 ÷2 ÷2 ÷2 ÷2 x<8 AND x> –3 That tells us x can be anything between 8 and – 3 . If asked for the integer values of x, that would be {7, 6, 5, 4, 3, 2, 1, 0, -1, -2} Now, CHECK your answers!!!! The 8 and – 3 should be the value of x that make the original an equation, not an inequality 2(8) − 5 = 11 and 2(-3) - 5 = 11 so that's good. What about numbers between them? 2(4) − 5 = 3 and 2(-1) - 5 = 7 good, they both came out less than 11 as they should. What about numbers outside that range? 2(20) − 5 = 35 and 2(-10) - 5 = 25 Great those were not supposed to work, and they didn't, they gave values outside the 8 to – 3 range!! Some problems will actually be easier to do starting by "guessing" right away, but only if you understand what you're doing and how absolute values work!!