Absolute Value Equations and Inequalities
Absolute value equations and inequalities involve expressions within absolute value symbols. When solving absolute value equations, the goal is to find the values of the variable that make the equation true. Absolute value inequalities are solved similarly, but the solution may involve a range of values rather than a single value.
6 Key excerpts on "Absolute Value Equations and Inequalities"
eBook - ePub- Stu Schwartz(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...CHAPTER 5 Equations and Inequalities CHAPTER 5 EQUATIONS AND INEQUALITIES Absolute Value Equations and Inequalities The definition of an absolute value of a number is its distance from zero. An absolute value is never a negative number. The formal definition of an absolute value is a piecewise function: For example, |8| = 8 and |−8| = −(−8) = 8. Absolute Value Equations From the above definition of absolute value, a linear equation involving absolute values can actually be treated as two equations. In the first equation, we replace the absolute value by a positive parenthesis. In the second equation, we replace the absolute value by a negative parenthesis. For example, | x + 2| = 7 means either (x + 2) = 7 or −(x + 2) = 7. We then solve both equations. We must check both solutions in the original equation because this solution process could introduce extraneous values. EXAMPLE Solve |2 x − 1| = 15. SOLUTION The two equations and their solutions are (2 x − 1) = 15 −(2 x − 1) = 15 2 x − 1 = 15 −2 x + 1 = 15 2 x = 16 −2 x = 14 x = 8 x = −7 Check with x = 8 : |16 − 1| = 15. Check with x = −7 : |−14 − 1| = 15. Both solutions check, and the answer is x = 8 or x = −7. EXAMPLE Solve |4 x − 5| + 5 x + 2 = 0. SOLUTION Form two equations as follows: 4 x − = + 5 x + 2 = 0 −(4 x − 5) + 5 x + 2 = 0 9 x = 3 −4 x + 5 +. 5 x + 2 = 0 x = −7 Check for, which does not equal zero. Check for x = −7: |4(−7) − 5| + 5(−7) + 2 = 33 − 35 + 2 =0. Therefore, only x = −7 is a solution. Absolute Value Inequalities For an inequality involving an absolute value, we replace the absolute value with a positive parenthesis and a negative parenthesis, just as we did for absolute value equations, being sure to preserve the original inequality sign. We then solve each inequality and place each solution on a number line to determine which intervals satisfy the inequality...
eBook - ePub- Mel Friedman(Author)
- 2012(Publication Date)
- Research & Education Association(Publisher)
...This value can be either positive or negative before the absolute value is taken. As a result, each absolute value equation actually contains two separate equations. When evaluating equations containing absolute values, proceed as follows: Example: | 5 – 3 x | = 7 is valid if either The solution set is therefore x = Remember, the absolute value of a number cannot be negative. So the equation | 5 x + 4 | = –3 would have no solution. INEQUALITIES The solution of a given inequality in one variable x consists of all values of x for which the inequality is true. A conditional inequality is an inequality whose validity depends on the values of the variables in the sentence. That is, certain values of the variables will make the sen tence true, and others will make it false. The sentence 3 – y > 3 + y is a conditional inequality for the set of real numbers, since it is true for any replacement less than 0 and false for all others, or y < 0 is the solution set. An absolute inequality for the set of real numbers means that for any real value for the variable, x, the sentence is always true. The sentence x + 5 > x + 2 is an absolute inequality because the expression on the left is greater than the expression on the right. A sentence is inconsistent if it is always false when its variables assume allowable values. The sentence x + 10 < x + 5 is inconsistent because the expression on the left side is always greater than the expression on the right side. The sentence 5 y < 2 y + y is inconsistent for the set of non-negative real numbers...
eBook - ePub- Mark Zegarelli(Author)
- 2011(Publication Date)
- For Dummies(Publisher)
...The value 3 is not included, because the second inequality is <. As a result, there should be an open circle at 3; thus, you can rule out Choice (E). Therefore, the correct answer is Choice (D). Evaluating a number’s absolute value The absolute value of a number is its value when you drop the minus sign. If the number doesn’t have a minus sign to begin with, its value stays the same. For example: |3| = 3 |–4| = 4 |0| = 0 Absolute value isn’t difficult, but you can avoid confusion by taking ACT questions with absolute value in two steps: First, evaluate and remove the absolute value bars, and then complete the problem. When working with absolute value, |–4| – |–6| = (A) 10 (B) 2 (C) 0 (D) –2 (E) –10 Begin by evaluating |–4| and |–6| separately to remove the absolute value bars: |–4| = 4 and |–6| = 6. Thus, you can rewrite the equation: |–4| – |–6| = 4 – 6 Now the problem becomes simple: 4 – 6 = –2, so the correct answer is Choice (D). In some cases, you may need to apply absolute value to a number line. Which of the following number lines expresses the set of all possible values of x for the inequality x ≥ |–2|? (F) (G) (H) (J) (K) To begin, evaluate the absolute value: |–2| = 2. So you can rewrite the inequality as follows: x ≥ 2 As you can see, the solution set includes 2 and all values greater than 2, so the correct answer is Choice (F). Understanding Factors and Multiples Two important concepts in arithmetic are factors and multiples. Both of these are related to the simple idea of divisibility: One positive integer is divisible by another if you can divide the first integer by the second without leaving a remainder. For example: 14 is divisible by 2, because 14 ÷ 2 = 7 (with no remainder) 14 is not divisible by 3, because 14 ÷ 3 = 4 r 2 (4 with a remainder of 2) When one number is divisible by another, you can describe the relationship between them using the words factor (the smaller number) and multiple (the larger number)...
eBook - ePub- Daniel Greenberg(Author)
- 2014(Publication Date)
- Research & Education Association(Publisher)
...First evaluate the quantity inside the brackets, and then make it a positive quantity. Thus, | 3 – 6 | = | –3 | = 3. 2. Since the quantity inside the brackets can be negative or positive and still have the same result (for example, | 4 | and | –4 | both equal 4), to solve an absolute value equation, set the quantity inside the brackets equal to a positive result as well as a negative result. 3. For example, | x + 5 | = 6 is solved by setting x + 5 = 6 and x + 5 = –6. The first (positive choice) yields x = 1 as an answer, but don’t forget the second (negative choice), which yields x = –11 as an answer. The answer is thus x = 1 and –11. 5.5 Problem 1: Solve: |4 x + 3| = 11 A) x = –2, x = 3.5 B) x = 2 C) x = 2, x = –3.5 D) x = –3.5 STRATEGY Set up two equations for the absolute value and solve each separately. THINK • |4 x + 3| = 11 is equivalent to 4 x + 3 = 11 and 4 x + 3 = –11. • Solve each equation to find that x = 2 and –3.5, making answer choice (C) the correct response. B. Inequalities 1. The signs for equality or inequality are: < less than > greater than = equal to ≤ less than or equal to ≥ greater than or equal to 2. Visualize the number line so the answers to inequality problems make sense. 3. Solve an inequality the same as you solve an equation, but replace the equal sign with the appropriate inequality sign. Thus, the values of x for the inequality x – 7 < 3 are found the same way as solving x – 7 = 3, except the < sign is used: This means values such as 9, 8, 0, –3, and all the numbers less than 10 are solutions to x – 7 < 3, but not x = 10. 4. The one exception to the simple rule above occurs when the solution involves multiplying or dividing by a negative number. In that case, the inequality sign reverses (because, for example, –7 is less, not more, than –3). For example, the inequality –3 x ≥ 6 is solved the same way as –3 x = 6, by dividing both sides by –3, but now the answer is x ≤ –2 (the inequality reverses direction)...
eBook - ePubMath Dictionary for Kids
The #1 Guide for Helping Kids With Math
- Theresa R. Fitzgerald(Author)
- 2021(Publication Date)
- Routledge(Publisher)
...ALGEBRAIC IDEAS DOI: 10.4324/9781003236443-4 Absolute Value The value of a number regardless of its sign, denoted by the numeral between two parallel lines. The distance from the origin to that number on a number line. Example: |+3| = 3 and |–3| = 3 Abundant Number A number in which the sum of its factors is larger than two times the number. Example: 12 = 1, 2, 3, 4, 6, 12 1 + 2 + 3 + 4 + 6 + 12 = 28 The sum of the factors is greater than 2 × 12, so 12 is an abundant number. The sum of the factors is greater than 2n, where n represents the original number. Additive Inverse The number that, when added to another number, yields zero. Example: +3 + −3 = 0 –3 is the additive inverse of +3. Balancing The process done in equations with an equal sign. An equation is balanced when both sides of the equal sign have the same amount, value, or mass. Examples: 7 = 3 + 4 6 − 1 = 5 These equations are balanced. 24 − 5 = 18 + 1 This equation is NOT balanced. This equation IS balanced. Base The number that is going to be raised to a power using an exponent. Example: Binomial A math expression that has two terms. Bi refers to two, and nomial means part, so a binomial is a polynomial with two terms, or two parts. It is the sum of two monomials. Binomials can have constants, variables, and exponents. They cannot be divided by a variable, have negative exponents, or have fractional exponents. Examples: 5 + 11 3x + 2y 3x 2 −2 5x + 3 2x + 4 6x 2 + 1 x 2 − 4x 6x + 3y Coefficient The numerical part of an algebraic term. The number used to multiply a variable in algebra. Examples: 3x 2 3 is the coefficient. 2y 2 is the coefficient. 5(a + b) 5 is the coefficient. 9x − 3 = 15 9 is the coefficient. Constant A fixed value or amount...
- Stu Schwartz(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...While there are other methods, the best way to accomplish this is to find the LCD (lowest common denominator) of all the fractions in the complex fraction. Multiply all terms by this LCD and you are left with a fraction that is magically no longer complex. EXAMPLE 17: Eliminate the complex fractions. a) b) c) SOLUTIONS: a) b) c) 13. Solving Fractional Equations Algebra has taught you that to simplify an expression such as, you need to find the LCD and multiply each fraction by this LCD in such a way that you have a common denominator. The LCD is x (x + 2), so you have. However, when you solve fractional equations (equations that involve fractions), you still find the LCD, but you multiply every term by the LCD. When you do that, all the fractions disappear, leaving you with an equation that is (hopefully) solvable. EXAMPLE 18: Solve SOLUTION: Note: Answers should be checked in the original equation because of the possibility of the denominator being zero. 14. Solving Absolute Value Equations Absolute value equations crop up in calculus, especially in BC calculus. The definition of the absolute value function is a piecewise function:. To solve an absolute value equation, split the absolute value equation into two equations, one with a positive sign in front of the parentheses and the other with a negative sign in front of the parentheses and solve each equation. EXAMPLE 19: Solve |2 x − 1| − x = 5. SOLUTION: EXAMPLE 20: Solve. SOLUTION: 15. Solving Inequalities You may think that solving inequalities is just a matter of replacing the equal sign with an inequality sign. In reality, they can be more difficult and are fraught with dangers. In calculus, expect to solve a number of inequalities on a regular basis. Solving inequalities is a simple matter if the inequalities are based on linear equations...
