Model 3: Rational Inequalities First: recall an important fact about inequalities. Consider the true inequality: 5 > 3. 15. Multiply both sides of the inequality by 2. Is it still true? 16. Multiply both sides of the inequality by *2. Is it still true? 17. In general5 when multiplying (or dividing) both sides of an inequality by a negative number5 what must be done to make sure the truth of the inequality stays the same? Now= our goal is to solve the following inequality. . . . . . . 2 . . If this were a rational equation: we could multiply both sides by the common dei'iominator1 :L' (:1: 3). However: since this is an inequality5 we would need to know whether we are multiplying both sides by a positive or negative value. Since the value of a: is unknown, the sign of 5132(1): 3) is unknown as well. InsteacL start by adding '3 to both sides of the inequality: so that zero is on the right side: :1: : 32::7+ 2 :52 153 20 18. Express the left side as a single fraction with a common denominator: then fully factor both numerator and denominator. 19. Rewrite the inequality using your fraction from #18. The right side should still just be zero. 2D. 21. 22. 23. 24. 25. 26. Label the zeros of both numerator and denominator on the number line below along with their multiplicities. These zeros split the arraxis into four intervals. m I TTL I TTL I Is the inequality from #19 true when the numerator is zero? If so, mark the zeros of the nul'nera.tor as solid points on your number line above. Otherwise, mark them as open points. Is the inequality from #19 true when the denominator is zero? If so, mark the zeros of the denominator as solid points on your number line above. Otherwise, mark them as open points. For each of the four intervals from #20, choose an xvalu: and plug it in to the fraction to see if it has a positive or LL :5 negative value on that interval. Label each interval on the number line with a \"+"' or . Note: The zeros you found are the endpoints of the intervals. Here, you are choosing :I:values from within, the intervals. The inequality from #19 is true when the fraction's value is either positive or zero. write the set of a: values for which the inequality is true as a union of intervals. Compare the signs on your number line with the multiplicities of the zeros. Complete these sentences. (a) To either side of a zero with multiplicity, the fractioan value has opposite signs. odd / even (b) To either side of a zero with multiplicity, the fractions value has the same sign. 0dr] fey-en Explain how the facts from #25 could have been used to ll out the sign chart without doing most of the :alculations in #23. Model 3: Rational Inequalities First, recall an important fact about inequalities. Consider the true inequality, 5 > 3- f 14. Multiply both sides or the inequality by 2. Is it still true? __ \" 15. Multiply both sides of the inequality by 2. Is it still true? , ust be done to 16. In general, when multiplying (or dividing) both sides of an inequality by a negative number, What m make sure the truth of the inequality stays the same? Now, our goal is to solve the following inequality. x2 'rc3 - 2 d , nsidering . If this were a rational equation, we could multiply both sides by the common denominator, :1: (1' 3) C0 #16, why would this strategy be problematic for the inequality? Hint: Is 22(x - 3) positive or negative? 18. Instead. start by adding 2 to both sides of the inequality, so that zero is on the right side. . :c _ 19. Next, express the left side as a single fraction with a common denominator. It)? 20. Fully factor both numerator and denominator and identify the zeros of each. Label them on the number line below along with their multiplicities. These numbers should split the r-axis into four intervals. signs zeros + mult. 21. Is the inequality true when the numerator is zero? If so, mark the zeros of the numerator as solid points on your number line above. Otherwise, mark them as open points. 22. Is the inequality true when the denominator is zero? If so, mark the zeros of the denominator as solid points on your number line above. Otherwise, mark them as open points. 23. Choose an r-value from one of the four intervals from #20, and plug it in to the fraction to see if it has a positive or negative value. Label that interval of the sign chart with a "+" or "-." 24. Use the multiplicities of the zeros to determine whether the sign of the fraction changes or does not change on either side of each zero. Use this information to fill in the rest of the signs. 25. The inequality is true when the fraction's value is either positive or zero. Write the set of x values for which the inequality is true as a union of intervals.Practice: Solving Rational Inequalities Solve the inequality. Graph the solution set on a number line and write the solution using interval notation. (Note: Your york must include number line reasoning.) 1. -1+5 50 2. 15 I+6 > 5