An Absolute Value Equation With One Solution

Absolute Value Equation with One Solution

Absolute value equations are fundamental in algebra, representing expressions where the variable appears within absolute value bars. Day to day, these equations can have zero, one, or two solutions depending on their structure. Which means an absolute value equation with one solution occurs when the expression inside the absolute value bars equals zero, or when the equation is structured such that only one value satisfies the condition. This unique solution happens when the absolute value expression is set equal to a non-negative number but the resulting equation after removing the absolute value has only one valid solution.

Understanding Absolute Value

The absolute value of a number represents its distance from zero on the number line, regardless of direction. For any real number x, the absolute value is denoted as |x| and defined as:

|x| = x, if x ≥ 0
|x| = -x, if x < 0

This definition creates two cases when solving equations involving absolute values. Even so, when we encounter |A| = B, where B is a constant, we must consider both possibilities: A = B or A = -B. Even so, for the equation to have exactly one solution, these two cases must either yield the same solution or one case must be invalid.

Conditions for a Single Solution

An absolute value equation |A| = B will have exactly one solution under the following conditions:

B equals zero: When the absolute value is set equal to zero, |A| = 0, the only solution occurs when A = 0. This is because absolute value represents distance, and the only number with zero distance from zero is zero itself.
One case is invalid: When solving |A| = B (with B > 0), we get A = B or A = -B. If one of these solutions makes the original equation undefined or contradicts the domain restrictions, only one valid solution remains.
Special equation structures: Some equations are constructed such that after removing the absolute value, one of the cases leads to a contradiction or no solution.

Solving Absolute Value Equations with One Solution

To solve an absolute value equation and determine if it has one solution, follow these steps:

Isolate the absolute value: Ensure the absolute value expression is alone on one side of the equation.
Set up two cases: Create two separate equations without absolute value bars:
- Case 1: Expression inside = value
- Case 2: Expression inside = -value
Solve both cases: Find potential solutions from both equations.
Check for extraneous solutions: Verify each solution in the original equation, as some may not satisfy the original equation due to the properties of absolute value.
Determine the number of solutions: Count the valid solutions. If only one solution works, you have an absolute value equation with one solution.

Examples of Absolute Value Equations with One Solution

Example 1: |x| = 0

This is the simplest case where the absolute value equals zero.

Solution: x = 0
Verification: |0| = 0, which is true.
Only one solution exists because zero is the only number whose absolute value is zero.

Example 2: |2x - 6| = 0

Step 1: Absolute value is already isolated.
Step 2: Set up one case (since B=0): 2x - 6 = 0
Step 3: Solve: 2x = 6 → x = 3
Step 4: Check: |2(3) - 6| = |0| = 0, which is true.
Only one solution exists.

Example 3: |x + 3| = -2

Step 1: Absolute value is isolated.
Step 2: Note that absolute value cannot be negative. The equation |x+3| = -2 has no solution because absolute value is always non-negative.
On the flip side, if we set up the cases:
- Case 1: x + 3 = -2 → x = -5
- Case 2: x + 3 = 2 → x = -1
But both solutions are invalid because substituting back gives | -5 + 3 | = |-2| = 2 ≠ -2 and | -1 + 3 | = |2| = 2 ≠ -2.
This equation has zero solutions, not one.

Example 4: |x - 4| + 2 = 2

Step 1: Isolate absolute value: |x - 4| = 0
Step 2: Set up one case: x - 4 = 0
Step 3: Solve: x = 4
Step 4: Check: |4 - 4| + 2 = 0 + 2 = 2, which is true.
Only one solution exists.

Example 5: |2x + 1| = x - 3

This equation has one solution because one case leads to a contradiction.

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Step 1: Absolute value is isolated.
Step 2: Set up two cases:
- Case 1: 2x + 1 = x - 3 → x = -4
- Case 2: 2x + 1 = -(x - 3) → 2x + 1 = -x + 3 → 3x = 2 → x = 2/3
Step 3: Check solutions:
- For x = -4: |2(-4) + 1| = |-8 + 1| = |-7| = 7, and x - 3 = -4 - 3 = -7. 7 ≠ -7, so invalid.
- For x = 2/3: |2(2/3) + 1| = |4/3 + 3/3| = |7/3| = 7/3, and x - 3 = 2/3 - 9/3 = -7/3. 7/3 ≠ -7/3, so invalid. Wait, both seem invalid? Let me recheck.

Actually, in Case 2, we have |2x+1| = x-3. But our solutions are x=-4 and x=2/3, both less than 3. So we require x-3 ≥ 0 → x ≥ 3. The right side must be non-negative because absolute value is always ≥0. So, no solution exists in this example.

Let me correct with a proper example:

Example 5 (corrected): |x + 2| = 3x - 6

Step 1: Absolute value is isolated.
Step 2: Set up two cases:
- Case 1: x + 2 = 3x - 6 → 2 + 6 = 3x - x → 8 = 2x → x = 4
- Case 2: x + 2 = -(3x - 6) → x + 2 = -3x + 6 → 4x = 4 → x = 1
Step 3: Check solutions:
- For x = 4: |4 + 2| = |6| = 6, and 3(4) - 6 = 12 - 6 = 6. 6 = 6, valid.
- For x = 1: |1 + 2| = |3| = 3, and 3(1) - 6 = 3 - 6 = -3. 3 ≠ -3, invalid.
Only x = 4 is valid, so one solution exists.

Scientific Explanation

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