Which number line represents the solutions to $|x+4|=2$?
Answer (1)
The absolute value equation ∣ x + 4∣ = 2 is split into two cases: x + 4 = 2 and x + 4 = − 2 .
Solving x + 4 = 2 gives x = − 2 .
Solving x + 4 = − 2 gives x = − 6 .
The solutions are x = − 6 and x = − 2 , which would be represented as two points on a number line: − 6 , − 2 .
Explanation
Understanding the Problem We are given the equation ∣ x + 4∣ = 2 and asked to find the number line that represents its solutions.
Solving the Equation To solve the absolute value equation ∣ x + 4∣ = 2 , we consider two cases:
Case 1: x + 4 = 2 . Subtracting 4 from both sides gives x = 2 − 4 = − 2 .
Case 2: x + 4 = − 2 . Subtracting 4 from both sides gives x = − 2 − 4 = − 6 .
Thus, the solutions are x = − 2 and x = − 6 .
Representing on a Number Line The number line representing the solutions will have closed circles (or dots) at -6 and -2.
Final Answer The solutions to the equation ∣ x + 4∣ = 2 are x = − 6 and x = − 2 .
Examples
Absolute value equations are useful in many real-world scenarios. For example, consider a machine that is supposed to fill bags with 100 grams of sugar. Due to manufacturing tolerances, the actual weight of sugar in a bag can vary by up to 2 grams. This situation can be modeled by the absolute value equation ∣ x − 100∣ = 2 , where x is the actual weight of sugar in the bag. Solving this equation helps determine the minimum and maximum possible weights of sugar in the bags, ensuring quality control.
