Numbers – Page 6 – JDM Educational

Absolute value appears often in algebra, and it is often used to represent distance. This raises the questions of how absolute value relates to negative numbers. So, can absolute value be negative? Absolute value cannot be negative – the output of absolute value is always positive or zero. However, we can use negative inputs for absolute value, and we can subtract inside of an absolute value. We can also get negative solutions for some absolute value equations and inequalities. Of course, there are some cases where an absolute value equation or inequality will have no solution, or where all of the solutions will be positive. It all depends on the numbers involved and the sign of the inequality, In this article, we’ll talk about absolute value and how they are connected to negative numbers. Let’s get going. Can Absolute Value Be Negative? Absolute value cannot be negative. That is, the output of the absolute value operator can never yield a negative number. The reason becomes clear if we look at the definition of absolute value. The absolute value (or modulus) of a real number x is denoted by |x| and it tells us the distance between 0 and x on the number line. Since distance can never be negative, it is also true that absolute value can never be negative. For example, |3| (in words, “the absolute value of 3”) is 3, since 3 is a distance of 3 units from 0 on the number line (3 units to the right of zero). Likewise, |-3| (in words, “the absolute value of -3”) is 3, since -3 is a distance of 3 units from 0 on the number line (3 units to the left of zero). If we ever have an equation with an absolute value that is equal to a negative number, then we have an equation with no solution. Why Is Absolute Value Never Negative? Absolute value is never negative because it tells us the magnitude of a number, not its direction with respect to zero. In other words, absolute value tells us how far away from zero a number is. It does not tell us the direction (whether the number is to the left or right of zero on the number line). There are three basic cases when we take the absolute value of a number x: x is negative – in this case, we drop the negative sign and take the positive value that remains. For example, the absolute value of -3 (denoted by |-3|) is 3.x is positive – in this case, we just take the value of x as the absolute value, since it is already positive. For example, the absolute value of 3 (denoted by |3|) is 3.x is zero – in this case, the absolute value of x is zero, since there is zero distance between 0 and x. That is, the absolute value of zero (denoted by |0|) is 0. SignOfInputSignOfOutputPositivePositiveZeroZeroNegativePositiveThis table summarizes the inputs and outputs for the absolute value function |x|. Is Absolute Value Always Positive? Absolute value is not always positive. It can also be zero, but it can never negative. The absolute value of zero is zero. The absolute value of any nonzero number is positive. Instead, we can use the expression “absolute value is always nonnegative”. This means that absolute value can be positive or zero, but not negative. How To Make Absolute Value Negative To make absolute value negative, you can “force” negation by putting a minus sign in front of (to the left of) the absolute value operator symbol. For example, -|3| (the negative of the absolute value of 3) will be – 3 (negative 3). Likewise, -|-3| (the negative of the absolute value of -3) will also be -3 (negative 3). Just as the absolute value of x (denoted by |x|) is always nonnegative, we can also say that the negative of the absolute value of x (denoted by -|x|) is always nonpositive (negative or zero). Can You Have A Negative In Absolute Value? You can have a negative in absolute value (that is, a negative number as an argument that you plug in to absolute value). This can happen in several ways. First, you can have a negative value as an argument to the absolute value operator. For example, |-3| = 3 has a negative in an absolute value. You can also subtract numbers within an absolute value. For example, |5 – 9| = |-4| = 4 has subtraction in an absolute value. Finally, you can have a variable expression with subtraction within an absolute value. For example, you can have an expression like |x – 2| or |x – y|, which tells you the distance between two values on the number line. You might also want to read about the concept of absolute difference, which tells us the distance between two values on the number line. Can An Absolute Value Equation Equal A Negative Number? An absolute value equation can equal a negative number, but there will be no solution in that case. Remember: an absolute value is nonnegative, meaning it is either zero or positive. The output of the absolute value operator is never negative. For example, the equation |x| = -2 has no solution. No matter which value of x you plug in, the result will always be at least 0 (for x = 0) or positive (for any nonzero value of x). Can The Solution Of An Absolute Value Equation Equal A Negative Number? The solution of an absolute value equation can equal a negative number. Here are some examples. Example 1: Absolute Value Equation With A Negative Solution Consider the absolute value equation |x + 4| = 5. Remember that the absolute value of 5 will equal 5, but the absolute value of -5 will also equal 5. So, in a sense, there are two “halves” to consider when we solve this equation: the positive and the negative. Then we really need to solve 2 separate equations: (A) x + 4 = 5(B) x + 4 = -5 Solving the first equation (A) gives us x = 1. Solving the second equation (B) gives us x = -9. In this particular example, one of the solutions to the absolute value equation is negative (x = -9), and the other is positive (x = 1). However, it is also possible for more than one solution of an absolute value equation to be negative. Example 2: Absolute Value Equation With Two Negative Solutions Consider the absolute value equation |x + 8| = 2. Again, there are two “halves” to consider when we solve this equation: the positive and the negative. We really need to solve 2 separate equations: (A) x + 8 = 2(B) x + 8 = -2 Solving the first equation (A) gives us x = -6. Solving the second equation (B) gives us x = -10. In this particular example, both of the solutions (x = -6 and x = -10) to the absolute value equation are negative. Example 3: Absolute Value Equation With Three Negative Solutions Consider the absolute value equation |x3 + 6x2 + 11x + 6| = 0. To solve this equation, we need to find out which x values make the expression inside the absolute value operator equal to zero. So, we want to solve x3 + 6x2 + 11x + 6 = 0. This factors as: (x+1)(x+2)(x+3) = 0 Which gives us solutions of x = -1, x = -2, and x = -3. Can An Absolute Value Inequality Be Negative? An absolute value inequality can be negative. However, there may or may not be solutions depending on the sign of the inequality. Can The Solution Of An Absolute Value Inequality Equal A Negative Number? The solution of an absolute value inequality can equal a negative number. An absolute value inequality can have negative solutions. Here are some examples. Example 1: Absolute Value Inequality With Negative Solutions Consider the absolute value inequality |2x – 3| > 9. To solve, we need to consider two separate inequalities: (A) 2x – 3 > 9(B) 2x – 3 12 or x > 6. There are no negative values for this half of the solution. Solving equation (B) gives us 2x < -6 or x < -3. All of the values are negative for this half of the solution. In fact, any values less than -3 (which are all negative) are solutions to this absolute value inequality. So, there are infinitely many negative solutions to this absolute value inequality. Example 2: Absolute Value Inequality With Negative Solutions Consider the absolute value inequality |3x – 6| < 12. To solve, we need to consider the chain of inequalities: -12 < 3x – 6 < 12 -6 < 3x < 18 -2 < x < 6 or (-2, 6). There are infinitely many negative values in the set (-2, 6) of solutions for the absolute value inequality. Conclusion Now you know how negative values and absolute value are connected. Although an absolute value never has an output that is negative, there can be negative solutions to an absolute value equation or inequality. Speaking of negative numbers, you might also want to read my article on negative expected value or check out my article on whether a normal distribution can be negative - there are some pitfalls to look out for! You can learn more about when to use absolute value in my article here. I hope you found this article helpful. If so, please share it with someone who can use the information. Don't forget to subscribe to my YouTube channel & get updates on new math videos! ~Jonathon