Problem 72 - Algebra and Trigonometry

Question

In Exercises 61–78, solve each absolute value equation or indicate that the equation has no solution. $$4\left|1-\frac{3}{4} x\right|+7=10$$

Step-by-Step Solution

Answer

The solution to the equation is \( x = -1, \frac{2}{3} \) .

Step 1: Isolate the Absolute Value Term

Subtract 7 from both sides of the equation to isolate the absolute value term on one side. This gives us \( 4\left|1-\frac{3}{4} x\right|=3 \) .

Step 2: Remove the Multiplier

Divide both sides of the equation by 4. This gives us the simplified equation \( \left|1-\frac{3}{4} x\right|= \frac{3}{4} \) .

Step 3: Consider Both Cases

Since an absolute value is always positive, we now have two separate equations to consider: \( 1-\frac{3}{4} x = \frac{3}{4} \) for the absolute value being positive, and \( 1-\frac{3}{4} x = -\frac{3}{4} \) for the absolute value being negative.

Step 4: Solve Each Case

Solving the first equation, subtract 1 from both sides to isolate \( x \). Then divide by \( -\frac{3}{4} \) to solve for \( x \). This gives \( x = -1 \).For the second equation, subtract 1 from both sides and then divide by \( -\frac{3}{4} \), giving \( x = \frac{2}{3} \) .

Key Concepts

Isolating Terms in Equations

To solve an absolute value equation, one of the first essential steps is to isolate the absolute value expression. In our example, the equation we start with is \(4\left|1-\frac{3}{4} x\right|+7=10\).
The goal here is to get \(\left|1-\frac{3}{4} x\right|\) by itself on one side of the equation. This means we need to remove any extra numbers surrounding it.

• First, eliminate the 7 by subtracting it from both sides: \(4\left|1-\frac{3}{4} x\right|=3\).
• Next, remove the multiplier 4 by dividing every term of the equation by 4: \(\left|1-\frac{3}{4} x\right|= \frac{3}{4}\).

This simplification helps us to focus only on the absolute value, paving the way for solving it effectively in the next steps.

Solving Linear Equations

After isolating the absolute value term, the next step is to solve the linear equations that arise from it. An absolute value equation like \(\left|1-\frac{3}{4} x\right|=\frac{3}{4}\) implies two potential cases because the absolute value expression can be either positive or negative.
You need to solve for \(x\) in both scenarios:

• First solve for when \(1-\frac{3}{4} x = \frac{3}{4}\).
• Second, solve for when \(1-\frac{3}{4} x = -\frac{3}{4}\).

These are straightforward linear equations.
In each equation, you perform similar steps:

• Subtract 1 from both sides to start isolating \(x\).
• Then, divide every term by \(-\frac{3}{4}\) to find \(x\).

This method ensures that the solution accounts for both cases generated by the absolute value expression.

Positive and Negative Cases

An absolute value equation has potential solutions in both positive and negative scenarios. Consider the expression \(\left|A\right| = B\). This means that \(A\) could either be \(B\) or \(-B\), and both are valid solutions.
Let's break down both cases for the problem:

• **Positive Case**: \( 1-\frac{3}{4} x = \frac{3}{4} \)
• **Negative Case**: \( 1-\frac{3}{4} x = -\frac{3}{4} \).

In many absolute value problems, considering both cases is crucial. Each case takes into account the essence of absolute values, which always result in non-negative outcomes. So, solving both positive and negative possibilities allows us to determine all potential \(x\) values that satisfy the initial equation.

Mathematical Operations

Mathematical operations are the building blocks for manipulating equations. In our problem, we applied several key operations to transform and solve the equation.
Here's a look at the operations involved:

• **Subtraction**: Initially used to isolate the absolute value term by removing the additive constant (subtracting 7 from both sides).
• **Division**: Employed to simplify the equation further by removing the multiplier 4, leading to \(\left|1-\frac{3}{4} x\right|=\frac{3}{4}\).
• **Arithmetic Manipulations**: In both cases, subtraction was used to further simplify the equation and isolate the \(x\) term.
• **Dividing Fractions**: Essential when solving for \(x\), dividing by each coefficient appropriately ensures a correct solution.

Understanding and applying these operations enables us to progress through the solution systematically, ensuring clarity and correctness at each step.