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2-5 and 2-6 Solving Inequalities With and Without Technology 1. The height, h, in metres, of a golf ball t seconds after it is hit can be modelled by the function h(t) = -4.9t2 + 32t + 0.2. When is the height of the ball 15 m or more? Clarify the problem: How is this question different than asking when the ball will be at 15m? ___________________ What does the graph of this function look like? ______________________________ How can 15m or more be written algebraically? ____________________________________ Use graphing technology to graph the function and solve the problem as accurately as the technology is able. The height of the ball is 15m or more when__________________________. Use the quadratic formula to find the exact solution. HW: Pg 129 #1 – 5, optional Pg 130 #6f, 7f, 8f, 9f, 11, 12, 14 mandatory 2. A rectangular in-ground swimming pool is to be installed. The engineer overseeing the construction project estimates that at least 1408 m3 of earth and rocks needs to be excavated. What are the minimum dimensions of the excavation if the depth must be 2 m more than one quarter of the width, and the length must be 12 m more than four times the width? Clarify the problem: How is this question different than asking when exactly 1408 cubic meters has to be excavated? _________________________________________ What does the graph of this function look like? __________________________ How can I write at least 1408 or more algebraically? ________________________________ Model this situation with an inequality Solve this problem algebraically. Remember that multiplying or dividing by a negative reverses the sense (direction) of the inequality. It’s often easier just to rearrange the inequality by adding and subtracting from both sides to avoid sense errors. Solve x3 + x2 – x + 1 ≤ 3x3 + 7x2 –13 x – 15 using the graph below (note different scales). Solve x3 + x2 – x + 1 ≤ 3x3 + 7x2 –13 x – 15 algebraically using intervals and by considering all cases Step 1: Rearrange to combine like terms on one side. Step 2: Factor to find roots. Step 3: Create interval table from -∞ to ∞ with the roots in order. Value of expression is zero at the roots. Use end behavior and/or pick a number between the roots and sub it in to determine when the expression is positive and when it is negative. The interval table has been started for you. The endpoints of the intervals have been shaded for clarity. Fill in all of the missing information. -∞ ≤ x ≤ Root1 Root3 ≤ x ≤ ∞ Root1 Root1 ≤ x ≤ Root2 Root2 Root2 ≤ x ≤ Root3 Root3 Interval -∞ ∞ (fill in roots) Sign of 0 0 0 0 expression Step 4: Communicate the solution to the inequality. HW: Pg 138 # 1 – 4, 11, 12 optional Pg 139 #5de, 6cd, 7cd, 8, 9 mandatory