What is the solution to the absolute value of 2x-5 = 21?

The equation “absolute value of 2x-5 = 21” asks us to find the values of ‘x’ that make the given statement true. In this case, we are dealing with an absolute value equation, which means the solution may have multiple possibilities. To find the solution, we need to understand the concept of absolute value and apply appropriate strategies.

Absolute value is a mathematical function that gives the magnitude or distance of a number from zero, ignoring its sign. It is denoted by two vertical lines enclosing the number, like |x|.

In the given equation |2x-5| = 21, we can rewrite it as two separate equations with their signs changed:
1. 2x-5 = 21: When the expression inside the absolute value bars is positive.
2. -(2x-5) = 21: When the expression inside the absolute value bars is negative.

Solving the first equation:
2x – 5 = 21
Adding 5 to both sides:
2x = 26
Dividing both sides by 2:
x = 13

Solving the second equation:
-(2x – 5) = 21
Distributing the negative sign:
-2x + 5 = 21
Subtracting 5 from both sides:
-2x = 16
Dividing both sides by -2:
x = -8

Therefore, the two solutions to the equation |2x-5| = 21 are x = 13 and x = -8.

FAQs:

1. What is an absolute value equation?

An absolute value equation is an equation that involves the absolute value of a variable. It requires finding the values of the variable that satisfy the equation while considering the possibility of negative and positive solutions.

2. How do I solve absolute value equations?

To solve an absolute value equation, you need to set up two separate equations, one with the expression inside the absolute value bars positive and one with it negative. Solve each equation separately and find any solutions that satisfy both.

3. Can an absolute value equation have multiple solutions?

Yes, an absolute value equation can have multiple solutions. This occurs when the absolute value expression on both sides of the equation evaluates to the same value or when there are two separate expressions inside the absolute value bars.

4. What does the absolute value symbol mean?

The absolute value symbol, represented by two vertical lines enclosing a number or variable, indicates the magnitude or distance of that number or variable from zero on the number line. It effectively makes the value positive, regardless of its original sign.

5. How do I know when to split an absolute value equation into two separate equations?

Splitting an absolute value equation into two separate equations is necessary when the expression inside the absolute value bars can evaluate to both positive and negative values. In such cases, solve each equation separately to cover all possibilities.

6. Are there any values of ‘x’ that could make the equation |2x-5| = 21 false?

No, there are no values of ‘x’ that could make the equation |2x-5| = 21 false, as we have accounted for all possibilities by considering the positive and negative solutions.

7. Can I check my solutions to verify if they are correct?

Yes, you can always substitute the found solutions for ‘x’ back into the original equation to check if they satisfy the equation. In this case, substituting x = 13 and x = -8 into |2x-5| = 21 yields true statements.

8. Are the two solutions the only possible solutions to the equation?

Yes, in this case, the solutions x = 13 and x = -8 are the only possible solutions for the equation |2x-5| = 21.

9. Can the absolute value of a number ever be negative?

No, the absolute value of a number can never be negative. It represents the distance from zero, which is always positive.

10. Can the absolute value of a variable be zero?

Yes, the absolute value of a variable can be zero. This occurs when the variable itself is zero.

11. Will an absolute value equation always have a solution?

No, not all absolute value equations have solutions. Some equations may result in no solution, depending on the given expression and the equation requirements.

12. Can the absolute value equation have more than two solutions?

Yes, an absolute value equation can have more than two solutions, especially when the absolute value expression on both sides of the equation can evaluate to the same value or when there are multiple expressions inside the absolute value bars. However, in this particular case, we have found two solutions.

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