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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:
 
Homework Assignment
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