Problem 50 - College Algebra with Modeling and Visualization

Question

Solve the equation symbolically. Then solve the related inequality. $$\left|\frac{1}{2} x-\frac{3}{4}\right|=\frac{7}{4}, \quad\left|\frac{1}{2^{x}}-\frac{3}{4}\right| \leq \frac{7}{4}$$

Step-by-Step Solution

Answer

Equation: \( x = 5 \) or \( x = -2 \). Inequality: \( x \geq 0 \).

Step 1: Understanding the Equation

We need to solve the equation \( \left| \frac{1}{2} x - \frac{3}{4} \right| = \frac{7}{4} \). The absolute value returns the distance from zero, leading us to two possible equations to solve: \( \frac{1}{2}x - \frac{3}{4} = \frac{7}{4} \) and \( \frac{1}{2}x - \frac{3}{4} = -\frac{7}{4} \).

Step 2: Solving the First Equation

First, solve \( \frac{1}{2} x - \frac{3}{4} = \frac{7}{4} \). Add \( \frac{3}{4} \) to both sides to get: \[ \frac{1}{2} x = \frac{7}{4} + \frac{3}{4} = \frac{10}{4} = \frac{5}{2} \] Multiply both sides by 2 to solve for \( x \): \[ x = 5 \]

Step 3: Solving the Second Equation

Next, solve \( \frac{1}{2} x - \frac{3}{4} = -\frac{7}{4} \). Add \( \frac{3}{4} \) to both sides to get: \[ \frac{1}{2} x = -\frac{7}{4} + \frac{3}{4} = -\frac{4}{4} = -1 \] Multiply both sides by 2 to solve for \( x \): \[ x = -2 \]

Step 4: Solutions to the Equation

The solutions to the equation \( \left| \frac{1}{2} x - \frac{3}{4} \right| = \frac{7}{4} \) are \( x = 5 \) and \( x = -2 \).

Step 5: Understanding the Inequality

Now solve the inequality \( \left| \frac{1}{2^{x}} - \frac{3}{4} \right| \leq \frac{7}{4} \). The inequality implies two sub-inequalities: \( -\frac{7}{4} \leq \frac{1}{2^{x}} - \frac{3}{4} \leq \frac{7}{4} \).

Step 6: Solving the First Inequality

Solve \( -\frac{7}{4} \leq \frac{1}{2^x} - \frac{3}{4} \). Add \( \frac{3}{4} \) to all sides:\[ -\frac{7}{4} + \frac{3}{4} \leq \frac{1}{2^x} \Rightarrow -1 \leq \frac{1}{2^x} \] Since \( \frac{1}{2^x} \) is always positive, this inequality is always true.

Step 7: Solving the Second Inequality

Solve \( \frac{1}{2^x} - \frac{3}{4} \leq \frac{7}{4} \). Add \( \frac{3}{4} \) to both sides:\[ \frac{1}{2^x} \leq \frac{7}{4} + \frac{3}{4} \Rightarrow \frac{1}{2^x} \leq \frac{10}{4} = \frac{5}{2} \]Since any \( x \geq 0 \) satisfies this, the solution set for the inequality is \( x \geq 0 \).

Step 8: Solutions to the Inequality

The solution to the inequality \( \left| \frac{1}{2^{x}} - \frac{3}{4} \right| \leq \frac{7}{4} \) is \( x \geq 0 \).

Key Concepts

Inequalities

Inequalities are mathematical expressions that show the relationship between two quantities where one is not strictly equal to the other. For example, the inequality \( \left| \frac{1}{2^{x}} - \frac{3}{4} \right| \leq \frac{7}{4} \) presents two conditions to consider:

• The expression can range between \(-\frac{7}{4}\) and \(\frac{7}{4}\), inclusive.
• The expression can be written as two sub-inequalities for clearer analysis: \( -\frac{7}{4} \leq \frac{1}{2^x} - \frac{3}{4} \leq \frac{7}{4} \).

Understanding inequalities involves graphing or analyzing these ranges on a number line to determine the solution set. Solutions are often expressed as intervals or ranges in inequality form indicating all the possible values that satisfy the original condition.

Symbolic Solving

Symbolic solving involves using algebraic manipulations to find the solution to an equation or inequality without necessarily evaluating it numerically right away. In symbolic solving, it's important to carefully handle each transformation of the equation or inequality to maintain equivalence.
For instance, in solving \( \left| \frac{1}{2} x - \frac{3}{4} \right| = \frac{7}{4} \), you first recognize that the absolute value denotes two possible cases: one where the expression inside is positive and another where it is negative:

• \( \frac{1}{2} x - \frac{3}{4} = \frac{7}{4} \)
• \( \frac{1}{2} x - \frac{3}{4} = -\frac{7}{4} \)

Solving each equation independently yields specific solutions, showing that each solution originates from the different possibilities inherent to absolute values.

Algebraic Expressions

Algebraic expressions comprise numbers, variables, and arithmetic operators (such as addition, subtraction) that define a specific mathematical relationship. In the given problem, expressions like \( \frac{1}{2} x - \frac{3}{4} \) are central to solving absolute value equations and inequalities.
When dealing with algebraic expressions in equations or inequalities:

• Identify key components like constants and coefficients.
• Use operations like addition or multiplication to isolate the variable.
• Rearrange the expressions to simplify the terms, making it easier to solve.

As demonstrated in the symbolic solving steps, understanding how to manipulate expressions is crucial for solving for unknown variables accurately. Such skills enhance one's ability to solve both simple and complex algebraic problems effectively.