Which equation has no solution?
[tex]$|4 x-2|=-6$[/tex]
[tex]$|-2-x|=9$[/tex]
[tex]$|3 x+6|=6$[/tex]
[tex]$|-2 x+8|=0$[/tex]
Answer (1)
Absolute value equations have no solution when the absolute value is equal to a negative number.
Examine each equation to see if the absolute value is equal to a negative number.
The equation ∣4 x − 2∣ = − 6 has no solution because the absolute value cannot be negative.
Therefore, the equation with no solution is ∣4 x − 2∣ = − 6 .
Explanation
Understanding the Problem We are given four equations involving absolute values, and we need to determine which one has no solution. Recall that the absolute value of any real number is non-negative. Therefore, an equation of the form ∣ a x + b ∣ = c has no solution if c < 0 .
Analyzing Each Equation Let's examine each equation:
∣4 x − 2∣ = − 6 . Since -6 < 0, this equation has no solution.
∣ − 2 − x ∣ = 9 . Since 9 > 0, this equation may have a solution.
∣3 x + 6∣ = 6 . Since 6 > 0, this equation may have a solution.
∣ − 2 x + 8∣ = 0 . Since 0 >= 0, this equation may have a solution.
Conclusion Therefore, the equation with no solution is ∣4 x − 2∣ = − 6 .
Examples
Absolute value equations are useful in real life for determining distances or deviations from a target value. For example, if you want to maintain the temperature of a room at 20 degrees Celsius, you might accept a deviation of up to 2 degrees. The equation ∣ T − 20∣ ≤ 2 describes the acceptable range of temperatures. Similarly, in manufacturing, absolute value equations help ensure that products meet specific size or weight tolerances. Understanding when these equations have no solution is crucial for setting realistic goals and expectations.
