Solving Inequalities Algebraically and Graphically

Objective:
In this lesson you learned how to solve linear inequalities,
inequalities involving absolute values, polynomial
inequalities, and rational inequalities .

Important Vocabulary Define each term or concept.

Solutions of an inequality All values of the variable for which the inequality is true.

Graph of an inequality The set of all points on the real number line that represent
the solution set of an inequality.
Linear inequality An inequality in one variable (usually x) that can be written in the
form ax + b < 0 or ax + b > 0,where a and b are real numbers with a ≠ 0.
Double inequality An inequality that represents two inequalities .

Critical numbers The x-values that make the polynomial in a polynomial inequality
equal to zero.
Test intervals Open intervals along the real number line in which the polynomial has
no sign changes .

I. Properties of Inequalities (Pages 54-55)	What you should learn How to recognize properties of inequalities
Solving an inequality in the variable x means . . . finding all the values of x for which the inequality is true. Numbers that are solutions of an inequality are said to satisfy the inequality. To solve a linear inequality in one variable, use the properties of inequalities to isolate the variable. When both sides of an inequality are multiplied or divided by a negative number, . . . the direction of the inequality symbol must be reversed. Two inequalities that have the same solution set are equivalent inequalities . Complete the list of Properties of Inequalities given below. 1) Transitive Property: a < b and b < c→a < c 2) Addition of Inequalities : a < b and c < d → a + c < b + d 3) Addition of a Constant c: a < b →a + c < b + c 4) Multiplication by a Constant c: For c > 0, a < b → ac < bc For c < 0, a < b → ac > bc
II. Solving a Linear Inequality (Pages 55-56)	What you should learn How to use properties of inequalities to solve linear inequalities
Describe the steps that would be necessary to solve the linear inequality 7x - 2 < 9x + 8 . Add 2 to each side. Subtract 9x from each side, and combine like terms . Divide each side by - 2 and reverse the inequality. Write the solution set as an interval. To use a graphing utility to solve the linear inequality 7x - 2 < 9x + 8 , . . . graph y₁ = 7x - 2 and y₂ = 9x + 8 in the same viewing window. Use the intersect feature of the graphing utility to find the point of intersection. Noticing where the graph of y₁ lies below the graph of y₂, write the solution set. The two inequalities - 10 < 3x and 14 ≥ 3x can be rewritten as the double inequality - 10 < 3x ≤ 14 .
III. Inequalities Involving Absolute Value (Page 57)	What you should learn How to solve inequalities involving absolute values
Let x be a variable or an algebraic expression and let a be a real number such that a ≥ 0. The solutions of \|x\| < a are all values of x that lie between - a and a . The solutions of \|x\| > a are all values of x that are less than - a or greater than a . Example 1: Solve the inequality: [- 15, - 7] The symbol is called a union symbol and is used to denote the combining of two sets Example 2: Write the following solution set using interval notation: x > 8 or x < 2
IV. Polynomial Inequalities (Pages 58-60)	What you should learn How to solve polynomial inequalities
Where can polynomials change signs ? Only at its zeros, the x-values that make the polynomial equal to zero. Between two consecutive zeros, a polynomial must be . . . entirely positive or entirely negative. When the real zeros of a polynomial are put in order, they divide the real number line into . . . intervals in which the polynomial has no sign changes. These zeros are the critical numbers of the inequality, and the resulting open intervals are the test intervals . Complete the following steps for determining the intervals on which the values of a polynomial are entirely negative or entirely positive: 1) Find all real zeros of the polynomial, and arrange the zeros in increasing order. The zeros of a polynomial are its critical numbers. 2) Use the critical numbers of the polynomial to determine its test intervals. 3) Choose one representative x-value in each test interval and evaluate the polynomial at that value. If the value of the polynomial is negative, the polynomial will have negative values for every x-value in the interval. If the value of the polynomial is positive, the polynomial will have positive values for every x-value in the interval. To approximate the solution of the polynomial inequality 3x² + 2x - 5 < 0 from a graph, . . . graph the associated polynomial y = 3x² + 2x - 5 and locate the portion of the graph that is below the x-axis. If a polynomial inequality is not given in general form, you should begin the solution process by . . . writing the inequality in general form—with the polynomial on one side and zero on the other side.
Example 3: Solve x² + x - 20 ≥ 0 . Example 4: Use a graph to solve the polynomial inequality - x² - 6x - 9 > 0 .
V. Rational Inequalities (Page 61)	What you should learn How to solve rational inequalities
To extend the concepts of critical numbers and test intervals to rational inequalities, use the fact that the value of a rational expression can change sign only at its zeros and its undefined values . These two types of numbers make up the critical numbers of a rational inequality. To solve a rational inequality, . . . first write the rational inequality in standard form. Then find the zeros and undefined values of the resulting rational expression. Form the appropriate test intervals and test a point from each interval in the inequality. Select the test intervals that satisfy the inequality as the solution set. Example 5:
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