What you should learn. The properties of inequalities. How to solve linear inequalities single inequality forms double inequality forms.
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1 P.6 Solving Inequalities What you should learn The properties of inequalities How to solve linear inequalities single inequality forms double inequality forms How solving inequalities can be used to determine the domain of even root radicals How to solve absolute value inequalities How to solve polynomial inequalities How to solve rational inequalities Aug 5 5:50 PM Aug 5 6:01 PM 1
2 P.6 Solving Inequalities What you should learn Some basic definitions relating to inequalities How to graph inequalities on the real number line How to express intervals on a real number line as inequalities The properties of inequalities Aug 5 5:50 PM Introduction Recall from Chapter P.1: <,, >, We already know: Expand our skills to solve the following: solve an inequality: Solutions are values of the variable that satisfy the inequality. Aug 9 8:40 AM 2
3 Solutions of Inequalities Solutions of inequalites may be shown as: intervals OR graphs of intervals Recall from Chapter P.1: Intervals may classified as : bounded OR unbounded Aug 9 8:40 AM Example 1 Intervals and Inequalities Write the inequality to represent each interval and state whether the interval is bounded or unbounded. a. b. c. d. Aug 9 11:34 AM 3
4 Properties of Inequalities Equivalent Inequalities: Aug 9 8:40 AM Oct 13 12:19 PM 4
5 Some Examples: Oct 13 12:24 PM SUMMARY: An equation is two expressions set equal to each other. Identity Equations are true for every value of the variable in the domain. Conditional Equations are not true for all or possibly any values of the variable in the domain. Equivalent Equations are different equations that have the same solution. Equivalent Equations can be generated by rearranging an equation using the Properties of Equality Aug 7 3:03 PM 5
6 P.6 Solving Inequalities What you should learn How to solve linear inequalities single inequality forms double inequality forms How solving inequalities can be used to determine the domain of even root radicals Aug 5 5:50 PM Linear Inequalities Strategy: Generate a sequence of equivalent inequalities using the... rules of algebra OR AND properties of inequalities until the variable is... isolated on the left (single inequality) isolated in the middle (double inequality) Aug 9 9:56 AM 6
7 Example 2 Solving Linear Inequalities Solve the inequality and graph the solution. a) Oct 18 11:47 AM Example 2 Solving Linear Inequalities Solve the inequality and graph the solution. b) Oct 18 11:47 AM 7
8 Example 3 Solving a Double Inequality Solve the inequality and graph the solution. Oct 18 11:47 AM Example 3 Solving a Double Inequality ALT: by parts Solve the inequality and graph the solution. Oct 18 11:47 AM 8
9 Finding the domain of an even root radical Find the domain of. Oct 18 11:54 AM SUMMARY: Strategy: Generate a sequence of equivalent inequalities using the... rules of algebra isolated on the left (single inequality) ORisolated in the middle (double inequality) ANDproperties of inequalities until the variable is... Solving inequalities can be used as a means for determining the domain of an even root radicals. Oct 18 1:04 PM 9
10 P.6 Solving Inequalities What you should learn How to solve absolute value inequalities Aug 5 5:50 PM Absolute Value Inequalities Strategy: Isolate absolute value using the... rules of algebra AND properties of inequalities Resolve the absolute value into two inequalities (or possible one double inequality) one tailed type two tailed type...as shown on the next slide Solve the two inequalities using previous methods Aug 9 9:56 AM 10
11 Oct 19 12:25 PM Example 4 Solving an Absolute Value Inequality Oct 19 12:31 PM 11
12 Example 4 Solving an Absolute Value Inequality Oct 19 12:31 PM How do we work backwards to identify the absolute value inequality from the graph of the solution? Oct 19 1:06 PM 12
13 Let's see if this strategy works for the other one... Oct 19 1:06 PM SUMMARY: Strategy: Isolate absolute value using the... rules of algebra AND properties of inequalities Resolve the absolute value into two inequalities (or possible one double inequality) one tailed type two tailed type Solve the two inequalities using previous methods Oct 19 12:43 PM 13
14 P.6 Solving Inequalities What you should learn How to solve polynomial inequalities Aug 5 5:50 PM Consider: Oct 20 10:52 AM 14
15 Polynomial Inequalities Strategy: Isolate the polynomial on the left side with only zero on the right side of the inquality using the... rules of algebra properties of inequalities Find all "critical points" of the polynomial where the polynonomial touches the x axis these are the only places the polynomial can change from being... ANDpositive to negative negative to positive Identify one convenient "test point" in each subinterval of the critical points Use the test point to identify the sign of the polynomial in that subinterval. Use the results of the previous step to write identify all of the subintervals that satisfy the inequality Aug 9 9:56 AM Example 5 Solving a Polynomial Inequality Solve: Oct 20 11:55 AM 15
16 Example 7 Finding the Domain of a Expression Find the domain of. Oct 20 11:55 AM SUMMARY: Strategy: Isolate the polynomial on the left side with only zero on the right side of the inquality using the... rules of algebra properties of inequalities Find all "critical points" of the polynomial where the polynonomial touches the x axis these are the only places the polynomial can change from being... ANDpositive to negative negative to positive Aug 9 9:56 AM 16
17 SUMMARY: Strategy (cont'd): Identify one convenient "test point" in each subinterval of the critical points Use the test point to identify the sign of the polynomial in that subinterval. Use the results of the previous step to write identify all of the subintervals that satisfy the inequality Aug 9 9:56 AM P.6 Solving Inequalities What you should learn How to solve rational inequalities Aug 5 5:50 PM 17
18 Consider the behavior of rational functions: Oct 20 10:41 AM Rational Inequalities Strategy: Move all terms to the left side of the inequality. Combine all terms on the left side of the inequality into a single term using common denominators Find all of the critical points... values of x where the rational expression is zero setting the numerator of the rational expression equal to zero values of x where the rational expression is undefined setting the denominator of the rational expression equal to zero Show all critical points on a number line. Aug 9 9:56 AM 18
19 Rational Inequalities Strategy: Identify one convenient test point in each subinterval critical points subinterval Use each test point to determine if the rational expression is positive or negative on each subinterval Identify all of the subintervals that satisfy the given rational inequality For < or >: For < or >: include the critical points corresponding to R(x) = 0. do not include the critical points corresponding to R(x) = 0. Never include the critical points corresponding to R(x) = undefined. Aug 9 9:56 AM Example 6 Solving a Rational Inequality Solve: Oct 20 2:41 PM 19
20 SUMMARY: Strategy: Move all terms to the left side of the inequality. Combine all terms on the left side of the inequality into a single term using common denominators Find all of the critical points... values of x where the rational expression is zero setting the numerator of the rational expression equal to zero values of x where the rational expression is undefined setting the denominator of the rational expression equal to zero Show all critical points on a number line. Aug 9 9:56 AM SUMMARY: Strategy: Identify one convenient test point in each subinterval critical points subinterval Use each test point to determine if the rational expression is positive or negative on each subinterval Identify all of the subintervals that satisfy the given rational inequality For < or >: For < or >: include the critical points corresponding to R(x) = 0. do not include the critical points corresponding to R(x) = 0. Never include the critical points corresponding to R(x) = undefined. Aug 9 9:56 AM 20
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