Problem 75 - Algebra 2

Question

Solve each equation. Check your solutions. \(|x+4|+3=17\)

Step-by-Step Solution

Answer

The solutions are \(x=10\) and \(x=-18\).

Step 1: Isolate the Absolute Value

We start by isolating the absolute value expression. From the equation \(|x+4|+3=17\), subtract 3 from both sides to obtain \(|x+4|=14\).

Step 2: Set Up Two Equations

The expression inside the absolute value can be either positive or negative, which leads us to two separate equations: \(x+4=14\) and \(x+4=-14\).

Step 3: Solve the First Equation

Solve the equation \(x+4=14\) by subtracting 4 from both sides, resulting in \(x=10\).

Step 4: Solve the Second Equation

Solve the equation \(x+4=-14\) by subtracting 4 from both sides, resulting in \(x=-18\).

Step 5: Verify Solutions

Substitute back into the original equation to verify both solutions. For \(x=10\), \(|10+4|+3=17\) simplifies to \(14+3=17\), which is true. For \(x=-18\), \(|-18+4|+3=17\) simplifies to \(14+3=17\), which is also true. Both solutions are valid.

Key Concepts

Isolation of Variables in Absolute Value Equations

To effectively solve absolute value equations, the first crucial step is to isolate the absolute value expression. In the equation \(|x+4|+3=17\), this means getting \(|x+4|\) by itself on one side of the equation. We do this by removing the constant added to the absolute value term, which is 3 in this case. Subtraction is the inverse operation to addition, so we subtract 3 from each side of the equation:

• Original: \(|x+4| + 3 = 17\)
• After subtracting 3: \(|x+4| = 14\)

Achieving this isolation helps set the stage for the ensuing steps, as the simplified form makes it's easier to address the nature of absolute values. It’s vital to perform this step correctly, as it directly affects the subsequent problem-solving processes that follow.

Solution Verification for Absolute Value Equations

Verification is an essential step to confirm that the obtained solutions indeed satisfy the original equation. Once you solve the absolute value equation, you should substitute your solutions back into the original equation to ensure their correctness. Let us consider the given equation \(|x+4|+3=17\) with solutions \(x=10\) and \(x=-18\).

• For \(x=10\): Substitute into the original: \(|10+4|+3=17\). This simplifies to \(14+3=17\), which confirms that \(x=10\) is a valid solution.
• For \(x=-18\): Substitute into the original: \(|-18+4|+3=17\). This simplifies to \(14+3=17\), validating \(x=-18\) as a solution as well.

This verification step is crucial for correct problem solving, as it checks for any mistakes in calculations and confirms the logic used. Ensuring that your solutions are accurate not only reassures the correctness but also builds confidence in algebraic manipulations.

Understanding Algebraic Equations Involving Absolute Values

Algebraic equations are at the heart of mathematical problem-solving and understanding them is crucial, especially when dealing with absolute values. An absolute value equation, like \(|x+4| = 14\), involves expressions where the absolute value needs to be solved for. This type of equation requires recognizing that the expression within the absolute value can be either positive or negative while still leading to the same positive result.To handle absolute value equations:

• Recognize that \(|x+4| = 14\) means \(x+4 = 14\) or \(x+4 = -14\).
• Create two separate linear equations based on this property.
• Solve each equation independently to find possible solutions for \(x\).

These steps leverage the concept that absolute values measure magnitude without regard to sign, presenting a unique yet straightforward challenge in algebra. Mastering the handling of such equations enhances overall algebraic skills and deepens understanding of mathematical concepts.