Problem 22 - Elementary and Intermediate Algebra

Question

Solve each equation. \(|x-7|=4\)

Step-by-Step Solution

Answer

The solutions are x = 3 and x = 11.

Step 1: Understand the Absolute Value

The absolute value equation \(|x-7|=4\) can be interpreted as the distance from \(x-7\) to 0 on the number line being 4.

Step 2: Set Up Two Potential Equations

Since the absolute value will balance out both positive and negative values, the equation \(|x-7|=4\) can be rewritten as two separate equations: \(x-7 = 4\) and \(x-7 = -4\).

Step 3: Solve the First Equation

Solve the first equation, \(x-7 = 4\), by adding 7 to both sides: \[x = 4 + 7 \]\[x = 11\].

Step 4: Solve the Second Equation

Solve the second equation, \(x-7 = -4\), by adding 7 to both sides: \[x = -4 + 7 \]\[x = 3\].

Step 5: Write the Solution Set

The solution set for \(|x-7|=4\) includes both values of \(x\) found in the previous steps: \(x = 3\) and \(x = 11\). Thus, the solutions are \([3, 11]\).

Key Concepts

Linear Equations

Let's dive into the exciting world of linear equations. Linear equations are one of the building blocks of algebra. A linear equation is any equation that forms a straight line when it is graphed. Its standard form is represented as \(ax + b = c\). Here, \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. Linear equations appear in different real-world contexts, such as calculating distance, finding rate, or determining a specific point in time. Understanding these helps us solve numerous practical problems. Here’s why understanding linear equations is essential:

• They represent simple relationships between variables.
• We can use them to predict values based on a given input.
• They form the basis for understanding more complex algebraic structures.

after getting a grasp on what they represent, solving them becomes a fun and valuable skill!

Step-by-Step Solutions

When tackling math problems, especially those involving absolute value, following a step-by-step process simplifies the task. Take our example, \(|x-7| = 4\): This involves understanding the concept of absolute value as the distance from zero. First, reframe the equation by considering both positive and negative possibilities. This leads to two equations: \(x-7 = 4\) and \(x-7 = -4\).Solving these structures methodically:

• For \(x-7 = 4\), add 7 to both sides to find \(x = 11\).
• For \(x-7 = -4\), add 7 to both sides to find \(x = 3\).

Once you master the step-by-step approach, equations become less intimidating and more manageable.

Algebraic Solutions

Algebraic solutions involve manipulating equations to find unknown values. The focus is on logically reducing and simplifying the problems using algebraic rules until we reach a solution. This process uses properties like addition, subtraction, and multiplication to isolate the variable.With absolute value equations like \(|x-7| = 4\), it's key to remember that this represents a distance on a number line. Thus, you use algebra to convert this into two simpler linear equations. We then solve these to determine all possible values of \(x\). Key points to finding algebraic solutions:

• Isolate the absolute value expression.
• Consider all potential solutions by splitting into two equations.
• Solve these equations by isolating \(x\).

Mastering algebraic solutions equips you with skills not only valuable for mathematics but applicable to solving real-life logical problems.