Absolute Value Reading A quick review of absolute value The absolute value of a number is its distance from zero on a number line. For example, the number 9 is 9 units from 0. Therefore its absolute value is 9. Negative numbers are more interesting, because the number -4 is still 4 units away from 0. The absolute value therefore is positive 4. The absolute value leaves a positive unchanged and makes a negative positive. An absolute value is written like this: x , and is read as “the absolute value of x”. Note: in certain calculator and computer programs, you may see it written as abs(x), which naturally means “the absolute value of x” but x is the accepted way to write it on your homework and tests. To force a number to be negative, you can write - x . This takes the number, makes it positive, and then takes the opposite of the number making it negative. Remember—just putting a negative sign in front of a number doesn’t make it negative—it makes it the opposite. Study these examples: 4 =4 −4 = 4 4+3 = 7 −4 − 3 = 7 3− 4 =1 − 4 = −4 − −4 = − 4 A More In-depth view of absolute Value The absolute value of an integer measure the magnitude (or size) of the integer, regardless of whether the integer is positive or negative. We disregard the sign or the directions of the number, and consider only the magnitude. The absolute value of any integer is always
positive. Thus the absolute value of -5 is 5, and the absolute value of 6 is 6, since absolute value is a distance and a distance is a positive quantity (or a nonnegative quantity, since it could be zero). When we consider points on a number line, the absolute value of a number is the distance between zero and the point representing the number, disregarding the direction of the number. The absolute value is also considered as a grouping symbol. This means that if the expression enclosed in the two vertical lines can be simplified, you must first do so before you take the absolute value of the result. The absolute value of x is defined as
x; if x ≥ 0 x = − x; if x < 0 The absolute value sign can be used in equations as well: −8 = x, thus x = 8 x = 8, thus x = 8 or x = −8. Re member that −8 is alos 8 so there are two solutions here ! x = −8, there are no solutions because the absolute value can never be negative. Absolute values are easy enough to compute when they contain constants, but all equations containing variable are more difficult. Suppose we are given the following equation x+2 =9 We can not assume that x+2 is positive or negative, so we can not simply "drop the bars." If x+2 were indeed negative, the absolute value of x+2 would really be -(x+2), since a negative times a negative equals a positive. We will solve using cases. The first case, or possibility, is that x+2 is positive. Taking the absolute value of a positive does not change the outcome. First Case: x + 2 = 9
The second case is that x+2 is negative. To get the absolute value of a negative, you have to negate it (which makes it positive again). Therefore |x+2| = -(x+2) Second Case: -(x + 2) = 9 Here we can solve both cases for x.
x+2 = 9 x=7
-(x+2) = 9 -x -2 = 9 -x = 11 x = -11
Our two solutions for |x+2|=9 are 7 and -11. Try them. They both work. More complicated equations can usually be solved the same way, by splitting the absolute value into two cases. You should check that you answers match the case, however. If you get a possible answer of 8 from the negative case, that can't be right. You should plug your answers back into the original equation to check for correctness. The statement x = 3 is graphed as two points x= -3 and x=3 because absolute value is considered as the distance from the origin and there are two points that are exactly 3 units from the origin.
Similarly, x = 5 indicates that the points are 5 units from the origin Solving absolute value equations using algebra notation. To solve absolute value equation using algebra we must first rewrite the given expression without the absolute value symbol using the following equivalent expressions. ax + b = p
is equivalent to ax+b=-p
Example 1; 3x + 5 = 4 First write two equations as represented above 3x+5= 4
Solve by isolating the variable 3x + 5 = 4 −5 = −5 3x = −1 1 x=− 3
3 x + 5 = −4 −5 = −5 3x = −9 x = −3
The two solutions can be graphed as shown below.
Use a separate sheet of paper to answer the following questions. These will be collected for an assessment grade tomorrow. 1. What does absolute value mean? 2. How many solutions will an absolute value equation have? How do you know? 3. We often talk about absolute value as distance from zero, what is another term used to describe absolute value? 4. Describe the graph of an absolute value equation. 5. How will you find absolute value shown on a calculator or in a computer program?