Chapter 2 Lesson 4 Solutions to Linear Inequalities

Linear Inequalities O A linear Inequality (or first-degree inequality) in the variable x is an inequality that can be written in the form ax + b > 0 where a ≠ 0 O The inequality symbol can be >, <, ≤, or ≥ O It is a linear inequality because the highest power of the variable is one.

Solutions to Linear Inequalities O Any value that can be substituted in for x and give a true inequality is part of the solution set for the inequality O i.e. 4x + 3 < 7x – 6 O 5 is in the solution set because: O 4(5) + 3 < 7(5) – 6 O < 35 – 6 O 23 < 29 O True

Steps for Solving Linear Inequalities O If linear inequality contains fractions with constant denominators, multiply both sides of the inequality by a positive number that will remove all denominators in the inequality. If there are two or more fractions, use the least common denominator (LCD) of the fractions. O Remove any parentheses by multiplication O Perform any additions/subtractions to get all terms containing the variable on one side and all other terms on the other side of the inequality. Combine like terms O Divide both sides of the inequality by the coefficient of the variable. Reverse the inequality symbol if this number is negative O Check the solution by substitution or with a graphing utility. If a real-world solution is desired, check the algebraic solution for reasonableness in the situation.

Examples

Profit O For a certain product, the respective weekly revenue and weekly cost are given by: O R(x) = 40x and C(x) = 20x where x is the number of units produced and sold. O For what levels of production will a profit result?

Body Temperature

Graphical Solutions to Linear Inequalities O There are two methods to solve linear inequalities with a graphing calculator O Intersection Method O X-intercept Method

Intersection Method Steps O Set the left side of the inequality equal to y1, set the right side equal to y2, and graph the equations using the calculator O Choose a viewing window that contains the point of intersection and find the point of intersection, with x- coordinate a. This is the value of x where y1=y2 O The values of x that satisfy the inequality represented by y1<y2 are those values for which the graph of y1 is below the graph of y2. O The values of x that satisfy the inequality represented by y1>y2 are those values for which the graph of y1 is above the graph of y2

Examples

X-Intercept Method Steps O Rewrite the inequality with all nonzero terms on one side of the inequality and combine like terms, getting f(x) > 0, f(x) < 0, f(x) ≥ 0, or f(x) ≤ 0 O Graph the nonzero side of the inequality (any window in which the x-intercept can be clearly seen is appropriate) O Find the x-intercept of the graph to find the solution to the equation f(x) = 0 (the exact solution can be found algebraically) O Use the graph to determine where the inequality is satisfied

Examples

Apparent Temperature O During the summer of 1998, Dallas TX endured 29 consecutive days where the temperature was at least 100°F. On many of those days, the combination of heat & humidity made it feel even hotter than it was. When the temperature is 100°F, the apparent temperature A (or heat index) depends on the humidity h (expressed as a decimal) according to A= h O For what humidity levels is the apparent temperature at least 110°F? O A ≥ 110 and A = h

Double Inequalities O A double inequality represents two inequalities connected by the word and or or O i.e. 5 < x < 9 O i.e. 4<x or x <-2 O Can be solved algebraically or graphically

Algebraic Steps to Solve Inequalities O Solve both parts of inequality separately, then put together at end

Examples

Compound Inequality O Three parts, with variable in middle O ‘And’ Inequality O Solve for variable in middle O Anything done to solve for variable needs to be done to all three parts of inequality

Examples

Solving Graphically O Graph all three parts to compound inequality O Find intersection points O Solution set is x values between intersection points

Examples

Course Grades O A student has taken four tests and has earned grades of 90%, 88%, 93%, and 85%. If all 5 tests count the same, what grade must the student earn on the last test so that the final course average is a B (that is, the average is at least 80% and less than 90%)

Expected Prison Time O The mean (expected) time y served in prison for a serious crime can be approximated by a function of the mean sentence length x, with y = 0.55x – 2.886, where x and y are measured in months. According to this model, how many months should a judge sentence a convicted criminal so that the criminal will be expected to serve between 37 and 78 months?

Homework O Pages O 1,5,9,13,17-19,21,24-26,28, 29,33,36,39,41,44,46, 49,53,56

Chapter Review O Pages O 2-50 even