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Compound Inequalities Section 2.6 Advanced Algebra 1 Warm-UP Solve and Graph: 1.) 2x - 4 ³ -20 +4 +4 2x ≥ -16 2 2 x ≥ -8 3.) 2.) -3x < 9 -3 x > -3 4x - 5 > -3x +16 +3x +3x 7x – 5 > 16 +5 +5 7x ≥ 21 7 7 -3 x>3 What integers are less than 5 and greater than 1? 2, 3, 4 What integers are greater than -2 and less than or equal to 3? -1, 0, 1, 2, 3 What integers are greater than -3 and less than 4? -2, -1, 0, 1, 2, 3 What integers are greater than 2 and less than 2? None! Sometimes we put more than one limit on the variable: Now x is still less than or equal to 9, but it must also be greater than or equal to –7. x 9 and x 7 If one number in a region completely satisfies an inequality, you can know that every number in that region satisfies the inequality. 15 -25 -20 -15 -10 -5 0 0 20 5 10 15 20 25 x 9 and x 7 “AND” problems first! Dealing with more than one inequality at a time. Find “x” that would satisfy both inequalities. Graph x ≥ 0 Graph x < 4 What if I wanted to graph “x ≥ 0 AND x < 4” on the same number line We can actually re-write this as: 0 ≤ x < 4 “x is between 0 and 4” We are looking for the overlapping section of both graphs up top GRAPH EACH INEQUALITY: 1.) -2 < y < 0 2.) 7 ≤ t ≤ 8 3.) 4 ≤ n ≤ 11 4.) x < 2 AND x < -3 Remember that “AND” means it has to satisfy both inequalities 5.) -2 < x + 2 ≤ 4 -2 -2 -2 -4 < x ≤ 2 7.) -5 ≤ 2x + 3 < 7 -3 -3 -3 6.) -6 ≤ 3x ≤ 12 3 3 3 -2 ≤ x ≤ 4 8.) -3 < -1 – 2x ≤ 5 -8 ≤ 2x < 4 2 2 2 +1 +1 +1 -2 < -2x ≤ 6 -2 - 2 -2 -4 ≤ x < 2 1 > x ≥ -3 -3 ≤ x < 1 One more: 9.) 1 ≤ -x – 6 ≤ 1 +6 +6 +6 7 ≤ -x ≤ 7 -1 - 1 -1 -7 ≥ x ≥ -7 -7 ≤ x ≤ -7 ON YOUR OWN: 1.) Write an compound inequality that represents the set of all real numbers greater or equal to -2 and less than 3. Then graph inequality. -2 ≤ x < 3 2.) Solve: -5 ≤ 2x + 3 < 7 -3 -3 -8 ≤ 2x < 4 2 2 2 -3 -4 ≤ x < 2 Now “OR” problems Dealing with more than one inequality at a time. Find “x” that would satisfy either inequality Warm-UP: Solve and Graph 1.) -7 < x – 4 < 3 +4 +4 -3 < x < 7 +4 2.) -3 ≤ 2x – 5 < 7 +5 +5 +5 2 ≤ 2x < 12 2 2 2 1≤x<6 What integers are less than 5 or greater than 1? All integers What integers are greater than -2 or greater than 3? All integers greater than -2 What integers are less than -3 or greater than 4? x < -3 or x > 4 What integers are greater than 2 or less than 2? All integers except 2 Graph x > 2 Graph x < -1 What if I wanted to graph “x > 2 OR x < -1” on the same number line We are looking for the UNION of all sections shaded Or means that a number only needs to meet one condition. Can you think of any numbers that satisfy one of these conditions? -25 -20 -15 -10 -5 0 5 10 15 20 x 12 or x 1 25 Graph each compound inequality: 1.) x < 0 or x > 5 3.) x – 4 ≤ 3 or 2x > 18 +4 +4 2 2 x≤7 OR x > 9 2.) x ≤ -1 or x ≥ 7 4.) 3x + 1 < 4 or 2x – 5 > 7 +5 +5 -1 -1 2x > 12 3x < 3 3 x<1 3 2 2 x>6 Look at the graph on the number line and create the inequality. First look at the boundary points. -25 -20 -15 -10 -5 0 5 10 15 15 x 20 20 25 Try this one: Again, begin by writing the boundary points: -25 -20 -15 -10 -5 0 5 10 15 20 x 5 or 10 x 25 Is there any one number that belongs to both shaded sections in the graph? NO! Say NO! x 5 or 10 x -25 -20 -15 -10 -5 0 5 10 15 20 25 Write the inequality from the graph: -25 -20 -15 -10 -5 0 5 10 15 10 x 5 20 25 Solve the inequality: 15 4 x 7 5 7 7 7 8 4 x 12 4 4 4 2 x 3 Final Answer is -3 x < 2 Solve and Graph these inequalities. 1.) -7 3-x < 5 2.) -25 < -5x < 0 3.) -3 -5-2x < 1 4.) 3 2x+3 7 5.) -6 -3x 12 6.) x-43 or 2x>18 7.) x-4<-8 or x+3>5 8.) 2x+31 or 3x-5>1 ANSWERS 1.) -2 < x 10 2.) 0 < x < 5 3.) -3 < x -1 4.) 0 x 2 5.) -4 x 2 6.) x 7 or x > 9 7.) x < -4 or x > 2 8.) x -1 or x > 2