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College Algebra Fifth Edition James Stewart Lothar Redlin Saleem Watson. Equations and Inequalities. 1. Chapter Overview. Equations are the basic mathematical tool for solving real-world problems. In this chapter , we learn How to solve equations
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College Algebra Fifth Edition James StewartLothar RedlinSaleem Watson
Chapter Overview • Equations are the basic mathematical tool for solving real-world problems. • In this chapter , we learn • How to solve equations • How to construct equations that real-life situations.
Basic Equations 1.1
Equations • An equation is a statement that two mathematical expressions are equal. • For example: 3 + 5 = 8
Variables • Most equations that we study in algebra contain variables, which are symbols (usually letters) that stand for numbers. • In the equation 4x + 7 = 19, the letter x is the variable. • We think of x as the “unknown” in the equation. • Our goal is to find the value of x that makes the equation true.
Solving the Equation • The values of the unknown that make the equation true are called the solutions or roots of the equation. • The process of finding the solutions is called solving the equation.
Equivalent Equations • Two equations with exactly the same solutions are called equivalent equations. • To solve an equation, we try to find a simpler, equivalent equation in which the variable stands alone on one side of the “equal” sign.
Properties of Equality • We use the following properties to solve an equation. • A, B, and C stand for any algebraic expressions. • The symbol means “is equivalent to.”
Properties of Equality • These properties require that you perform the same operation on both sides of an equation when solving it. • So, if we say “add –7” when solving an equation, that is just a short way of saying “add –7 to each side of the equation.”
Linear Equation • The simplest type of equation is a linear equation, or first-degree equation. • It is an equation in which each term is either a constant or a nonzero multiple of the variable.
Linear Equation—Definition • A linear equation in one variable is an equation equivalent to one of the form • ax + b = 0 • where: • a and b are real numbers. • x is the variable.
Linear Equations • These examples illustrate the difference between linear and nonlinear equations.
E.g. 1—Solving a Linear Equation • Solve the equation • 7x – 4 = 3x + 8 • We solve this by changing it to an equivalent equation with all terms that have the variable x on one side and all constant terms on the other.
E.g. 1—Solving a Linear Equation • 7x – 4 = 3x + 8 • (7x – 4) + 4 = (3x + 8) + 4 (Add 4) • 7x = 3x + 12 (Simplify) • 7x – 3x = (3x + 12) – 3x(Subtract 3x) • 4x = 12 (Simplify) • ¼ . 4x = ¼ . 12 (Multiply by ¼) • x = 3 (Simplify)
Checking Your Answer • It is important to check your answer. • We do so in many examples. • In these checks, LHS stands for “left-hand side” and RHS stands for “right-hand side” of the original equation.
Checking Your Answer • We check the answer of Example 1. • x = 3: • LHS = 7(3) – 4 = 17 • RHS = 3(3) + 8 = 17 • LHS = RHS
E.g. 2—Solving an Equation Involving Fractions • Solve the equation • The LCD of the denominators 6, 3, and 4 is 12. • So, we first multiply each side of the equation by 12 to clear denominators.
An Equation that Simplifies to a Linear Equation • In the next example, we solve an equation the doesn’t look like a linear equation, but it simplifies to one when we multiply by the LCD.
E.g. 3—Equation Involving Fractional Expressions • Solve the equation • The LCD of the fractional expressions is(x + 1)(x – 2) = x2 – x – 2. • So, as long as x≠ –1 and x≠ 2, we can multiply both sides by the LCD.
Extraneous Solutions • It is always important to check your answer. • Even if you never make a mistake in your calculations. • This is because you sometimes end up with extraneous solutions.
Extraneous Solutions • Extraneous solutions are potential solutions that do not satisfy the original equation. • The next example shows how this can happen.
E.g. 4—An Equation with No Solution • Solve the equation • First, we multiply each side by the common denominator, which is x – 4.
E.g. 4—An Equation with No Solution • Now, we try to substitute x = 4 back into the original equation. • We would be dividing by 0, which is impossible. • So this equation has no solution.
Extraneous Solutions • The first step in the preceding solution, multiplying by x – 4, had the effect of multiplying by 0. • Do you see why? • Multiplying each side of an equation by an expression that contains the variable may introduce extraneous solutions. • That is why it is important to check every answer.
Power Equations • Linear equations have variables only to the first power. • Now let’s consider some equations that involve squares, cubes, and other powers of the variable. • Such equations will be studied more extensively in Sections 1.3 and 1.5.
Power Equation • Here we just consider basic equations that can be simplified into the form • Xn = a • Equations of this form are called power equations • They are solved by taking radicals of both sides to the equation.
Solving a Power Equation • The power equation Xn = a has the solution • X = if n is odd • X = if n is even and a≥ 0 • If n is even and a < 0, the equation has no real solution.
Examples of Solving a Power Equation • Here are some examples of solving power equations. • The equation x5 = 32 has only one real solution: x = = 2. • The equation x4 = 16 has two real solutions: x = = ±2.
Examples of Solving a Power Equation • Here are some examples of solving power equations. • The equation x5 = –32 has only one real solution: x = = –2. • The equation x4 = –16 has no real solutions because does not exist.
E.g. 5—Solving Power Equations • Solve each equation. • x2 – 5 = 0 • (x – 4)2 = 5
Example (a) E.g. 5—Solving Power Equations • x2 – 5 = 0 • x2 = –5 • x = ± • The solutions are x = and x = .
Example (b) E.g. 5—Solving Power Equations • We can take the square root of each side of this equation as well. • (x – 4)2 = 5 • x – 4 = ± • x = 4 ± (Add 4)
E.g. 6—Solving Power Equations • Find all real solutions for each equation. • x3 = – 8 • 16x4 = 81
Example (a) E.g. 6—Solving Power Equations • Since each real number has exactly one real cube root, we can solve this equation by taking the cube root of each side. • (x3)1/3 = (–8)1/3 • x = –2
Example (b) E.g. 6—Solving Power Equations • Here we must remember that if n is even, then every positive real number has two real nth roots. • A positive one and a negative one. • If n is even, the equation xn = c (c > 0) has two solutions, x = c1/n and x = –c1/n.
Example (b) E.g. 6—Solving Power Equations
Equations with Fractional Power • The next example shows how to solve an equation that involves a fractional power of the variable.
E.g. 7—An Equation with a Fractional Power • Solve the equation • 5x2/3 – 2 = 43 • The idea is to first isolate the term with the fractional exponent, then raise both sides of the equation to the reciprocal of that exponent. • If n is even, the equation xn/m = c has two solutions, x = cm/n and x = –cm/n.
E.g. 7—An Equation with a Fractional Power • 5x2/3 – 2 = 43 • 5x2/3 = 45 • x2/3 = 9 • x = ±93/2 • x = ±27
Several Variables • Many formulas in the sciences involve several variables. • It is often necessary to express one of the variables in terms of the others. • In the next example, we solve for a variable in Newton’s Law of Gravity.
E.g. 8—Solving for One Variable in Terms of Others • Solve for the variable M in • The equation involves more than one variable. • Still, we solve it as usual by isolating M on one side and treating the other variables as we would numbers.
E.g. 9—Solving for One Variable in Terms of Others • The surface area A of the closed rectangular box shown can be calculated from the length l, the width w, and the height h according to the formula:A = 2lw + 2wh + 2lh • Solve for w in terms of the other variables in the equation.
