Problem 25 - Algebra for College Students

Question

Solve each equation. $$ \left|1+\frac{3}{4} x\right|=7 $$

Step-by-Step Solution

Answer

The solutions are \( x = 8 \) and \( x = -\frac{32}{3} \).

Step 1 - Understand Absolute Value

Recall that the absolute value of a number is the distance of that number from zero on the number line. This means that \(|A| = B\) implies \(A = B\ or \ A = -B\).

Step 2 - Set Up Two Equations

Using the property of absolute value, set up two separate equations: \( 1 + \frac{3}{4} x = 7 \) and \( 1 + \frac{3}{4} x = -7 \).

Step 3 - Solve the First Equation

Solve \( 1 + \frac{3}{4} x = 7 \) by isolating \( x \):1. Subtract 1 from both sides: \(\frac{3}{4}x = 6\).2. Multiply both sides by \( \frac{4}{3} \): \( x = \frac{4 \cdot 6}{3}=8 \).

Step 4 - Solve the Second Equation

Solve \( 1 + \frac{3}{4} x = -7 \) similarly:1. Subtract 1 from both sides: \(\frac{3}{4}x = -8\).2. Multiply both sides by \( \frac{4}{3} \): \( x = \frac{4 \cdot (-8)}{3} = -\frac{32}{3} \).

Key Concepts

absolute value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. For example, the absolute value of both 7 and -7 is 7. This is because they are both 7 units away from zero. This can be written as \(|7| = 7\) and \(|-7| = 7\).
When solving equations involving absolute values, like \(|A| = B\), we need to consider both scenarios: \(A = B\) and \(A = -B\).
In this particular problem, we started with \(|1+\frac{3}{4}x|=7\), so we break it into two separate equations: \(1+\frac{3}{4}x=7\) and \(1+\frac{3}{4}x=-7\). This approach allows us to address both possible cases.

solving linear equations

Solving linear equations involves finding the value of the variable that makes the equation true. Let's focus on the two separate equations from our absolute value problem:

• \(1+\frac{3}{4}x=7\)
• \(1+\frac{3}{4}x=-7\)

To solve each equation, we isolate the variable on one side of the equation. For example:
For \(1+\frac{3}{4}x=7\), we subtract 1 from both sides to get \(\frac{3}{4}x=6\). Then, we multiply both sides by the reciprocal of the fraction \( \frac{4}{3} \), making the variable x alone: \( x = \frac{24}{3} \), which simplifies to 8.
For \(1+\frac{3}{4}x=-7\), we again subtract 1 to obtain \(\frac{3}{4}x=-8\). When we multiply both sides by \(\frac{4}{3}\), we get \( x = -\frac{32}{3} \). Linear equations like these are straightforward once you isolate the variable.

algebraic manipulation

Algebraic manipulation involves performing operations to both sides of an equation to keep it balanced. These operations can include addition, subtraction, multiplication, or division. In our example, we performed two main operations:

• Subtraction: We subtracted 1 from both sides of the equations \(1 + \frac{3}{4}x =7\) and \(1 + \frac{3}{4}x = -7\) to simplify them.
• Multiplication by the reciprocal: Since \( \frac{3}{4}x = 6\) and \( \frac{3}{4}x = -8\), we used the reciprocal of \(\frac{3}{4}\) which is \(\frac{4}{3}\) to make the coefficient of x equal to 1.

By keeping both sides of the equation balanced through each step, we ensure the solution remains valid. Algebraic manipulation empowers us to move terms around and simplify equations efficiently.