$3|x+4|+12=9$
If you have multiple answers, separate them with a comma. Do not include any spaces. If there are no solutions type "no solutions".
Answer (1)
Subtract 12 from both sides: 3∣ x + 4∣ = − 3 .
Divide both sides by 3: ∣ x + 4∣ = − 1 .
Since the absolute value cannot be negative, there are no solutions.
Final answer: n oso l u t i o n s
Explanation
Problem Analysis We are given the equation 3∣ x + 4∣ + 12 = 9 . Our goal is to solve for x .
Isolating the Absolute Value Term First, we subtract 12 from both sides of the equation:
3∣ x + 4∣ + 12 − 12 = 9 − 12
3∣ x + 4∣ = − 3
Dividing by 3 Next, we divide both sides of the equation by 3:
3 3∣ x + 4∣ = 3 − 3
∣ x + 4∣ = − 1
Analyzing the Result The absolute value of any real number is always non-negative. Therefore, ∣ x + 4∣ cannot be equal to -1. This means there is no solution to the given equation.
Final Answer Therefore, the equation 3∣ x + 4∣ + 12 = 9 has no solutions.
Examples
Absolute value equations can be used to model real-world situations involving distances or deviations from a target value. For example, in manufacturing, the acceptable deviation from a specified length of a part can be modeled using an absolute value equation. If the specified length is 10 cm and the acceptable deviation is 0.1 cm, the actual length x must satisfy ∣ x − 10∣ ≤ 0.1 . Solving this inequality gives the range of acceptable lengths for the part. Similarly, in finance, absolute value can represent the difference between an investment's actual return and its expected return.
