Problem 1 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-50: Solve the equation. $$ |x+4|=11 $$

Step-by-Step Solution

Answer

The solutions are \(x = 7\) and \(x = -15\).

Step 1: Understanding Absolute Value Equation

The equation given is \(|x+4| = 11\). An absolute value equation \(|A| = B\) can have two possible solutions: \(A = B\) or \(A = -B\), as the absolute value represents distance, which can be in two opposite directions on a number line.

Step 2: Solve for Positive Case

For the equation \(|x+4| = 11\), start by solving the positive case: \[x+4 = 11\] Subtract 4 from both sides: \[x = 11 - 4\] \[x = 7\] This is the first solution.

Step 3: Solve for Negative Case

Now, solve the negative case:\[x+4 = -11\] Subtract 4 from both sides: \[x = -11 - 4\] \[x = -15\]This is the second solution.

Step 4: Verify Solutions

Substitute \(x = 7\) back into the original equation:\(|7+4| = |11| = 11\), which is correct.Now substitute \(x = -15\):\(|-15+4| = |-11| = 11\), which is also correct.Both solutions satisfy the original equation.

Key Concepts

Solving Equations

To solve an equation like \(|x+4| = 11\), it's important to understand the nature of absolute value. Absolute value refers to the distance of a number from zero on the number line, irrespective of direction. When solving absolute value equations, we consider two scenarios because both a positive and a negative number can have an equivalent absolute value.
The first step is to set the expression inside the absolute values equal to 11 (i.e., without the absolute value symbol):

• \(x + 4 = 11\)
• Subtract 4 from both sides to isolate \(x\): \(x = 7\)

For the second scenario, set the expression equal to -11 to account for the negative side:

• \(x + 4 = -11\)
• Subtract 4 from both sides to achieve: \(x = -15\)

The solutions \(x = 7\) and \(x = -15\) make the original equation valid, as both answers satisfy \(|x+4| = 11\).

Algebra

Algebra is the branch of mathematics that deals with symbols and rules for manipulating those symbols. In this particular exercise, the key algebraic concept is working with absolute value equations. Understanding how to handle absolute values and apply algebraic rules is crucial.

When solving such equations, we primarily focus on isolating the variable. This exercise exemplifies two vital algebraic operations:

• Using the properties of equality to manipulate the equation: adding, subtracting, multiplying, or dividing both sides to simplify.
• Recognizing that each absolute value equation can potentially yield two linear equations. Solving both of these provides a complete set of possible solutions.

These operations build a foundation that aids in solving more complex algebraic problems. Studying absolute value equations like \(|x+4| = 11\) helps in enhancing logical reasoning and problem-solving skills in mathematics.

Mathematics Education

In mathematics education, having a solid understanding of absolute value equations is critical for developing a comprehensive skill set in algebra. Absolute value, which measures how far a number is from zero without regard to direction, is central to high-school and college-level algebra courses.

Teaching these concepts involves focusing on:

• Conceptual understanding: Appreciating why an absolute value equation \(|A| = B\) leads to two potential solutions \(A = B\) and \(A = -B\).
• Problem-solving techniques: Developing strategies to systematically address both cases in an absolute value equation.
• Application to real-world scenarios: Absolute values are useful when discussing distances, errors, or fluctuations, thus giving context to abstract mathematical concepts.

Understanding these principles allows students to solve absolute value equations effectively while supporting their broader mathematical education journey.