Problem 62 - Algebra 2

Question

Solve each equation. Check your solutions. \(|x-3|=17\)

Step-by-Step Solution

Answer

The solutions are \(x = 20\) and \(x = -14\).

Step 1: Understanding Absolute Value

The given equation is \(|x - 3| = 17\). We know that an absolute value equation \(|A| = B\) implies two possible equations: \(A = B\) or \(A = -B\). This is because the absolute value of any number is always positive.

Step 2: Splitting into Two Equations

Apply the rule of absolute values to the equation \(|x - 3| = 17\). This results in two separate equations:1. \(x - 3 = 17\)2. \(x - 3 = -17\).

Step 3: Solving the First Equation

For the first equation: \(x - 3 = 17\).- Add 3 to both sides to isolate \(x\).- \(x = 17 + 3 = 20\).

Step 4: Solving the Second Equation

For the second equation: \(x - 3 = -17\).- Add 3 to both sides to isolate \(x\).- \(x = -17 + 3 = -14\).

Step 5: Verifying the Solutions

Verify both solutions by substituting back into the original equation \(|x - 3| = 17\).- For \(x = 20\): \(|20 - 3| = |17| = 17\), which is correct.- For \(x = -14\): \(|-14 - 3| = |-17| = 17\), which is correct.Both solutions satisfy the original equation.

Key Concepts

Solving Equations

Solving equations involving absolute values can initially seem tricky, but once you understand the basic principle, it becomes straightforward. When faced with an equation like \(|x - 3| = 17\), it's essential to recognize that the absolute value function translates to two distinct potential scenarios. This is because absolute value, by definition, measures the distance from zero, and thus is always non-negative. As a result, when \(|x - 3| = 17\), it implies two possible equations: \(x - 3 = 17\) and \(x - 3 = -17\). Both must be solved independently to identify all potential solutions. By addressing each equation separately, you ensure that you account for both the positive and negative solutions of the absolute value.

Algebraic Expressions

Algebraic expressions are fundamentally about representing numbers with symbols and variables to model real-world phenomena. In the context of our example, the expression \(|x - 3|\) captures a specific relationship. When tackling algebraic equations, it's crucial to work systematically. Begin with isolating the variable of interest. In the first equation \(x - 3 = 17\), you add 3 to both sides to isolate \(x\). This results in \(x = 20\). Similarly, for the second equation \(x - 3 = -17\), you again add 3 to both sides, leading to \(x = -14\). These steps highlight the importance of performing the same operations on both sides of an equation to maintain equality. Consistency in handling variables and constants during operations ensures accuracy and helps clarify the path to the correct solution.

Verification of Solutions

Verification is a crucial part of solving equations to ensure that the solutions are indeed correct. After solving the absolute value equation \(|x - 3| = 17\), you must substitute your solutions back into the original equation. This step verifies the correctness of your solutions. For \(x = 20\), plugging it back in gives \(|20 - 3| = 17\), which checks out as true. Similarly, substituting \(x = -14\) yields \(|-14 - 3| = |-17| = 17\). Both checks confirm that the solutions are valid. Verification not only validates your answers but reinforces understanding of how absolute values and equations work together. Always make it a point to verify your answers as a final step to ensure accuracy and gain confidence in your problem-solving skills.