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Purdue MA 15200 - Solving Absolute Value Inequalities

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1Summer MA 15200 Lesson 14 Section 1.7 (part 2) and Sections 1.1 & 2.8 I Solving Absolute Value Inequalities Absolute Value Inequalities: or , if 0uc uc c<≤≥ The inequality uc<indicates all values less than c units from the origin. Therefore uc< is equivalent to the compound inequality cuc−<<. There is a similar statement for uc≤ . Absolute Value Inequalities: , , if 0ucuc c>≥ > The inequality uc> indicates all values more than c units from the origin. Therefore uc> is equivalent to the inequality statement or ucuc<−>. There is a similar statement for uc≥. To help you keep the two cases straight in your head, I recommend thinking of a number line. If the absolute value is greater than a positive number c, it is greater than that many units away from zero. If the absolute value is less than a positive number c, it is within that many units of zero. Ex 1: Solve each. Write solutions using interval notation and graph the solutions on a number line. Hint: Always isolate the absolute value before writing an inequality without the absolute value. ) 4 6ax+< c−cc c−u-ccuu-c c2 ) 3 5 1 952) 13bxxc++≤+< Ex 2: Solve each inequality. Write the solutions using interval notation and graph the solutions on a number line. ) 2 5) 3 72) 3 4 2 7axxbcx−>+≥−+>3II Applied Problems Ex 3: Mary wants to spend less than $600 for a DVD recorder and some DVDs. If the recorder of her choice costs $425 and DVDs cost $7.50 each, how many DVDs could Mary buy? Ex 4: The percentage, P, of US voters who used punch cards or lever machines in national elections can be modeled by the formula 2.5 63.1Px=−+ where x is the number of years after 1994. In which years will fewer than 35.7% of US voters use punch cards of lever machines? Ex 5: A college provides its employees with a choice of two medical plans shown in the following table. Plan 1: $100 deductible payment 30% of the remaining payments Plan 2: $200 deductible payment 20% of the remaining payments For what size hospital bills is plan 2 better for the employee than plan 1? (Assume the bill is over $200.)4Ex 6: The room temperature in a public courthouse during a year satisfies the inequality 371 <−T where T is in degrees F. Express the range of temperatures without the absolute value symbol. III Rectangular Coordinate System Rectangular Coordinate System: Every point on a rectangular coordinate system is represented by an ordered pair , (x, y). x and y are called the coordinates of the point. Ex 7: a) In which quadrant(s) do the coordinates of a point have the same sign? b) In which quadrant would the point (-2, 3) be found? c) The point (12, 0) is found on which axis? d) What point would be 5 right and 6 down from the origin? IV Graphs of Equations An equation in 2 variables can be represented on a rectangular coordinate system by plotting points (ordered pairs) that satisfies the equation. The complete graph contains all ordered pairs whose coordinates satisfy the equation. 1. Make a table by selecting a value for x and solving for y (or vice-versa). 2. Plot enough points to be able to sketch a smooth curve or line to represent the graph. y-axis originI II III IV x-axis There are 4 quadrants, represented by Roman Numerals counter-clockwise from the upper right.5Ex 8: Graph the following equations. Use the values of 3, 2, 1,0,1,2 and 3−−− for x. 2)3ay x=− 12)byx= x y 22 xy x y 22 xy6 312)1cy x=− + V Intercepts An x-intercept of a graph is an x-coordinate of a point where the graph intersects the x-axis. The y-intercept of a graph is the y-coordinate of a point where the graph intersects the y-axis. Yes, our textbook uses single numbers to describe intersepts. Some textbooks use the complete ordered pair to define the intercepts. (See the study tip at the bottom of page 94 of the textbook.) Since an intercept always lies on an axis, the coordinate other than the one given as the x-intercept or y-intercept is zero. Zero is your friend! Ex 9: Use the following graphs to identify the intercepts. a) x y 22 xy 22 xy7 b) c) d) ()0,4− ()4,0 ()0,2− ()3,2−()0,1x (0,5)8VI Applied Problems On page 96 of your textbook is a graph showing the probability of divorce by the wife’s age at the time of marriage. The two mathematical models that approximate the data displayed in the graphs are 45dn=+ (where d = the percentage of marriages ending in divorce for n years after the marriage for a wife under 18 at time of marriage) and 2.3 1.5dn=+ (for a wife over 25 at time of marriage). Ex 10: a) Use the graph to approximate the percentage of marriages ending in divorce after 5 years of marriage if the wife was under 18. b) Use the correct model (formula) to approximate the percentage of marriages ending in divorce after 5 years of marriage


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