1. Isolate the radical term by adding x to each side; then square both sides of the equation.
2. Set the equation equal to 0, simplify, and then factor it.
Two solutions appear, x = –2 or x = –3.
3. Check for an extraneous solution.
Substituting the solutions, you see that both numbers work in this case.
1. Solve for x: .
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2. Solve for x: .
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3. Solve for x: .
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4. Solve for x: .
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5. Solve for x: .
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6. Solve for x: .
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Doubling the Fun with Radical Equations
Some radical equations have two or more radical terms. When you have more than one radical term, solving the equation takes more than one squaring process because, after you square both sides, you still have a radical term. Therefore, to solve a radical equation with multiple radical terms, you square the two sides first, then isolate the remaining radical term on one side, and then square both sides again.
In general, squaring a binomial (the sum or difference of two terms) is easier if only one of the two terms is a radical term, so a good technique is to rewrite the equations, putting a radical term on each side before you perform the first squaring process.
Q. Solve for x: .
A. x = –2
1. Subtract from each side (doing so places a radical on each side of the equation).
2. Square both sides, simplify the terms, and get the remaining radical term on one side and all the other terms on the opposite side.
3. Each of the terms is divisible by 4. Divide every term by 4 and then square both sides.
4. Set the quadratic equation equal to 0, factor it, and solve for the solutions to that equation.
5. Check the two solutions.
When you test each solution in the original equation, you find that –2 is a solution but 1 is an extraneous root.
Q. Solve for x: .
A. x = 0
1. Square both sides. (Be sure to square the 2 in front of the radical before distributing over the other terms.) Simplify and isolate the radical on one side.
2. Square both sides.
3. Set the quadratic equation equal to 0, factor it, and solve for the solutions to that equation.
4. Check the two solutions.
When you test each solution in the original equation you find that 0 is a solution but is an extraneous root.
7. Solve for x: .
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8. Solve for x: .
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Solving Absolute Value Equations
An absolute value equation contains the absolute value operation. Seems rather obvious, doesn’t it? When solving an absolute value equation, you have to change its form to solve it (just as you do with radical equations; refer to the preceding sections). When solving an absolute value equation in the form , take the following steps:
1. Rewrite the original equation as two separate equations and solve the two equations separately for two different answers.
The two equations to solve are ax + b = c or ax + b = –c
2. Check the results in the original equation to ensure the answers work.
Generally, both answers work, but you need to check the results to be sure the original equation didn’t have a nonsense statement (like having a positive equal to a negative) in it.
Q. Solve for x: .
A. x = or x = –2. Rewrite the absolute value equation as two different equations: 4x + 5 = 3 or 4x + 5 = –3. Solve 4x + 5 = 3, which gives . Solve 4x + 5 = –3, which gives x = –2.
Checking the solutions:
and .
Q. Solve for x: .
A. x = 3 or x = –9. Before applying the rule to change the absolute value into linear equations, add 5 to each side of the equation. This gets the absolute value by itself, on the left side: .
Now the two equations are 3 + x = 6 and 3 + x = –6. The solutions are x = 3 and x = –9, respectively. Checking these answers in the original equation, and .
9. Solve for x: .
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10. Solve for y: .
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11. Solve for w: .
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12. Solve for y: .
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13. Solve for x: .
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14. Solve for y: .
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Answers to Problems on Radical and Absolute Value Equations
This section provides the answers (in bold) to the practice problems in this chapter.
1. Solve for x: . The answer is x = 39. First square both sides and then solve for x by adding 3 to each side:
Then check:
2. Solve for x: . The answer is . First square both sides and then subtract 9 from each side. Find the square root of each side and check to see whether the answers work:
Then check:
3. Solve for x: . The answer is . First move x to the right side:
Check: . So 4 isn’t a solution; it’s extraneous.
Check: . So is the only solution.
4. Solve for x: . The answer is x = 12. First subtract 9 from each side and then square both sides. Set the quadratic equal to 0 to factor and solve for x:
Using the multiplication property of zero (see Chapter 13), you have x – 12 = 0, x = 12 or x – 7 = 0, x = 7.
5. Solve for x: . The answer is both .
First add 7 to each side. Then square both sides, set the equation equal to 0, and solve for x:
Using the multiplication property of zero (see Chapter 13), you have x + 7 = 0, x = –7 or x + 6 = 0, x = –6.
6. Solve for x: .The answer is both . First square both sides of the equation; then set it equal to 0 and factor:
7. Solve for x: . The answer is . First square both sides of the equation. Then keep the radical term on the left and subtract x and 9 from each side. Before squaring both sides again, divide by 6:
Square both sides of the new equation:
Check:
8. Solve for x: . The answer is . First move a radical term to the right, square both sides, simplify, and, finally, isolate the radical term on the right. You can then divide each side by 5:
Square both sides again, set the equation equal to 0, and factor:
Using the multiplication property of zero (see Chapter 13), you have x – 8 = 0, x = 8.
Check:
9. Solve for x: |x – 2| = 6. The answer is . First remove the absolute value symbol by setting what’s inside equal to both positive and negative 6. Then solve the two linear equations that can be formed:
Check:
10. Solve for y: |3y + 2| = 4. The answer is . First remove the absolute value symbol by setting what’s inside equal to both positive and negative 4. Then solve the two linear equations that can be formed:
Check:
11. Solve for w: . The answer is . First subtract 3 from each side. Then remove the absolute value symbol by setting what’s inside equal to both positive and negative 3:
The two linear equations that are formed give you two different answers:
12. Solve for y: . The answer is . First subtract 2 from each side. Then divide each side by 3:
Then rewrite without the absolute value symbol by setting the expression inside the absolute value equal to positive or negative 2: . Then simplify the resulting linear equations:
13. Solve for x: . The answer is . First rewrite the equation without the absolute value symbol:
14. Solve for y: . This equation doesn’t have an answer. Here’s why:
Note that is impossible because can never equal a negative number.
Chapter 16
Getting Even with Inequalities
In This Chapter
Playing by the rules when dealing with inequalities
Solving linear and quadratic inequalities
Getting the m
ost out of absolute value inequalities
Working with compound inequalities
An inequality is a mathematical statement that says that some expression is bigger or smaller than another expression. Sometimes the inequality also includes an equal sign with the inequality sign to show that you want something bigger than or equal to, or smaller than or equal to.
The good news about solving inequalities is that nearly all the rules are the same as solving an equation — with one big difference. The difference in applying rules comes in when you’re multiplying or dividing both sides of an inequality by negative numbers. If you pay attention to what you’re doing, you shouldn’t have a problem.
This chapter covers everything from basic inequalities and linear equalities to the more challenging quadratic, absolute value, and complex inequalities. Take a deep breath. I offer you plenty of practice problems so you can work out any kinks.
Using the Rules to Work on Inequality Statements
Working with inequalities really isn’t that difficult if you just keep a few rules in mind. The following rules deal with inequalities (assume that c is some number):
If a > b, then adding c to each side or subtracting c from each side doesn’t change the sense (direction of the inequality), and you get .
If a > b, then multiplying or dividing each side by a positive c doesn’t change the sense, and you get or .
If a > b, then multiplying or dividing each side by a negative c does change the sense (reverses the direction), and you get or .
If a > b, then reversing the terms reverses the sense, and you get b < a.
Q. Starting with –20 < 7, perform the following operations: Add 5 to each side, multiply each side by –2, change the number order, and then divide each side by 6.
A. –4 < 5
Adding 5 didn’t change the sense.
Multiplying by –2 reverses the sense.
Flip-flopping the terms to put the numbers in order from smaller to larger reverses the sense.
Dividing by a positive number does nothing to the direction of the sense.
1. Starting with 5 > 2, add 4 to each side and then divide each side by –3; simplify the result.
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2. Starting with , multiply each side by –4; then divide each side by –2 and simplify the result.
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Rewriting Inequalities by Using Interval Notation
Interval notation is an alternate form for writing inequality statements. Interval notation uses brackets and parentheses instead of greater than or less than signs. Many books in higher mathematics courses use interval notation; I show it here so you’ll be acquainted with the notation. In general, you use parentheses to indicate that the number is not included in the statement and brackets to show that it is included (with greater than or equal to and less than or equal to signs). I show you several examples of interval notation versus inequality notation.
Q. Write and , using interval notation.
A. and
becomes . The numbers that x represents have no boundary as they get smaller, so is used. The 6 is included, and a bracket indicates that.
becomes .The –4 isn’t included, so a parenthesis is used. You always use a parenthesis with infinity or negative infinity because there’s no end number.
Q. Write and , using interval notation.
A. and
becomes . The number –5 isn’t included, so a parenthesis is used. The 3 is included, so a bracket is used.
becomes (4,7). Neither 4 nor 7 are included, so parentheses are used. A caution here: The interval (4,7) looks like the coordinates for the point (4,7) in graphing. You have to be clear (use some actual words) to convey what you’re trying to write when you use the double parentheses for intervals.
3. Write , using interval notation.
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4. Write , using interval notation.
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Solving Linear Inequalities
Solving linear inequalities involves pretty much the same process as solving linear equations: Move all the variable terms to one side and the number to the other side. Then multiply or divide to solve for the variable. The tricky part is when you multiply or divide by a negative number. Because this special situation doesn’t happen frequently, people tend to forget it. Remember: If you multiply both sides of –x < –3 by –1, the inequality becomes x > 3; you have to reverse the sense.
Another type of linear inequality has the linear expression sandwiched between two numbers, like . The main rule here is that whatever you do to one section of the inequality, you do to all the others. For more on this, go to “Solving Complex Inequalities,” later in this chapter.
Q. Solve for x: .
A. or . Subtract 6x from each side and subtract 5 from each side; then divide by –9:
Note that you can do this problem another way to avoid division by a negative number. See the next example for the alternate method.
Q. Solve for x:
A. or . Add 3x to each side and subtract 14 from each side. Then divide by 9. This is the same answer, if you reverse the inequality and the numbers.
5. Solve for y: .
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6. Solve for x: 3(x + 2) > 4x + 5.
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7. Solve for x: .
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8. Solve for x: .
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Solving Quadratic Inequalities
When an inequality involves a quadratic expression, you have to resort to a completely different type of process to solve it than that used for linear inequalities. The quickest and most efficient method to find a solution is to use a number line.
After finding the critical numbers (where the expression changes from positive to negative or vice versa), you use a number line and place some + and – signs to indicate what’s happening to the factors. (See Chapter 1 for some background info on the number line and where numbers fall on it.)
Q. Solve the inequality x2 – x > 12.
A. x < –3, x > 4 or
1. Subtract 12 from each side to set it greater than 0.
2. Factor the quadratic and determine the zeros. (These are the critical numbers.)
You want to find out what values of x make the different factors equal to 0 so that you can separate intervals on the number line into positive and negative portions for each factor.
The two critical numbers are 4 and –3; they’re the numbers that make the expression equal to 0.
3. Mark these two numbers on the number line.
4. Look to the left of the –3, between the –3 and 4, and then to the right of the 4 and indicate, on the number line, what the signs of the factors are in those intervals.
Algebra I Workbook For Dummies Page 14