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ОглавлениеChapter 2
Solving Some Equations and Inequalities
The object of solving equations and inequalities is to discover which number or numbers will create a true statement in the given expression. The main techniques you use to find such solutions include factoring, applying the multiplication property of zero, creating sign lines, finding common denominators, and squaring both sides of an equation. Your challenge is to determine which techniques work in a particular problem and whether you have a correct solution after applying those techniques.
The Problems You’ll Work On
In this chapter, you’ll work with equations and inequalities in the following ways:
Writing inequality solutions using both inequality notation and interval notation
Solving linear and quadratic inequalities using a sign line
Determining the solutions of absolute value equations and inequalities
Taking on radical equations and checking for extraneous roots
Rationalizing denominators as a method for finding solutions
What to Watch Out For
Keep in mind that when solving equations and inequalities, your challenges will include
Factoring trinomials correctly when solving quadratic equations and inequalities
Choosing the smallest exponent when the choices include fractions and negative numbers
Assigning the correct signs in intervals on the sign line
Recognizing the solution when a factor of the expression is x
Determining which solutions are viable and which are extraneous
Squaring binomials correctly when working with radical equations
Finding the correct format for a binomial’s conjugate
Using Interval and Inequality Notation
61–70 Write the expression using the indicated notation.
61. Write the expression in interval notation.
62. Write the expression in interval notation.
63. Write the interval notation as an inequality expression.
64. Write the interval notation as an inequality expression.
65. Describe the graph using inequality notation.
Illustration by Thomson Digital
66. Describe the graph using inequality notation.
Illustration by Thomson Digital
67. Describe the graph using inequality notation.
Illustration by Thomson Digital
68. Describe the graph using interval notation.
Illustration by Thomson Digital
69. Describe the graph using interval notation.
Illustration by Thomson Digital
70. Describe the graph using interval notation.
Illustration by Thomson Digital
Solving Linear Inequalities
71–74 Solve the inequality for x.
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Solving Quadratic Inequalities
75–80 Solve the nonlinear inequality for x.
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Solving Absolute Value Inequalities
81–85 Solve the absolute value inequality for x.
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Working with Radicals and Fractional Notation
86–89 Change the expression using rules for fractional exponents.
86. Rewrite the radical expression with a fractional exponent.
87. Rewrite the radical expression with a fractional exponent.
88. Rewrite the fractional exponent as a radical expression.
89. Rewrite the fractional exponent as a radical expression.
Performing Operations Using Fractional Exponents
90–94 Simplify the expression.
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Factoring Using Fractional Notation
95–100 Factor the greatest common factor (GCF) from each term and write the factored form.
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Solving Radical Equations
101–110 Solve for x. Check for any extraneous solutions.
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Rationalizing Denominators
111–120 Simplify by rationalizing the denominator.
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