The absolute value of an expression represents the distance of that expression from zero on the number line. In this case, we have the absolute value of 2x-5 equal to 21. To find the solution, we need to examine two cases: when the expression inside the absolute value is positive and when it is negative.
The solution to the absolute value of 2x-5 = 21 is x = 13 or x = -8.
Let’s break down the solution into two cases:
Case 1: 2x-5 is positive
When 2x-5 is positive, the absolute value of 2x-5 is equal to 2x-5. Setting 2x-5 equal to 21, we can solve for x:
2x-5 = 21
2x = 26
x = 13
So, when 2x-5 is positive (greater than zero), the value of x is 13.
Case 2: 2x-5 is negative
When 2x-5 is negative, the absolute value of 2x-5 is equal to -(2x-5) or -2x+5. Setting -2x+5 equal to 21, we can solve for x:
-2x+5 = 21
-2x = 16
x = -8
Therefore, when 2x-5 is negative (less than zero), the value of x is -8.
Frequently Asked Questions:
1. What is an absolute value?
An absolute value represents the distance of a number or expression from zero on the number line.
2. How do I solve an equation with absolute value?
To solve an equation with absolute value, you need to consider both cases: when the expression inside the absolute value is positive and when it is negative.
3. What does it mean if an equation has no solutions?
If an equation has no solutions, it means that there is no value or values that satisfy the equation. In other words, there is no possible value for the variable that makes the equation true.
4. Can an absolute value be negative?
No, an absolute value is always positive or zero.
5. What is the absolute value of a negative number?
The absolute value of a negative number is the positive value of that number. For example, the absolute value of -8 is 8.
6. Can an absolute value be zero?
Yes, the absolute value of zero is zero itself.
7. What is the purpose of absolute value in mathematics?
The purpose of absolute value in mathematics is to represent the magnitude or distance of a number or expression from zero, without considering its sign.
8. Can an absolute value equation have more than one solution?
Yes, an absolute value equation can have more than one solution, as in the case of the equation 2|x| = 4, which has x = 2 and x = -2 as its solutions.
9. How can I check if my solution is correct?
You can check if your solution is correct by substituting the value of the variable back into the original equation and verifying that it makes the equation true.
10. Is it necessary to check both cases when solving an absolute value equation?
Yes, it is necessary to check both cases because the absolute value equation may have solutions in both situations: when the expression inside the absolute value is positive and when it is negative.
11. Are there any alternative methods to solve absolute value equations?
There are alternative methods to solve absolute value equations, such as using a graphing calculator or applying the definition of absolute value as a piecewise function.
12. Can an absolute value equation have no solution?
Yes, an absolute value equation can have no solution if there is no value for the variable that satisfies the equation.