Problem 18 - Precalculus Student Solutions Manual 5th

Question

Solve each equation. $$\left|\frac{2 x+3}{3 x-4}\right|=1$$

Step-by-Step Solution

Answer

The solutions are \( x = 7 \) and \( x = 0.2 \).

Step 1 - Understand the Absolute Value Equation

The given equation is an absolute value equation: \[ \left|\frac{2x+3}{3x-4}\right|=1 \]The absolute value expression \( |A| = 1 \) means that A could be either 1 or -1.

Step 2 - Set Up Two Separate Equations

Since \( \left| \frac{2x+3}{3x-4} \right| = 1 \) can be true if either \[ \frac{2x+3}{3x-4} = 1 \] or \[ \frac{2x+3}{3x-4} = -1 \], set up these two equations separately.

Step 3 - Solve the First Equation

Solve the equation \( \frac{2x+3}{3x-4} = 1 \):1. Multiply both sides by \( 3x-4 \):\( 2x+3 = 3x-4 \).2. Subtract \( 2x \) from both sides:\( 3 = x - 4 \).3. Add 4 to both sides:\( x = 7 \).

Step 4 - Solve the Second Equation

Solve the equation \( \frac{2x+3}{3x-4} = -1 \):1. Multiply both sides by \( 3x-4 \):\( 2x+3 = -3x+4 \).2. Add \( 3x \) to both sides:\( 5x+3 = 4 \).3. Subtract 3 from both sides:\( 5x = 1 \).4. Divide both sides by 5:\( x = \frac{1}{5} \) or \( x = 0.2 \).

Step 5 - Verify the Solutions

The solutions found are \( x = 7 \) and \( x = 0.2 \). Verify by checking if they satisfy the original equation:1. For \( x=7 \):\( \left| \frac{2(7) + 3}{3(7) - 4} \right| = \left| \frac{17}{17} \right| = 1 \).2. For \( x=0.2 \):\( \left| \frac{2(0.2) + 3}{3(0.2) - 4} \right| = \left| \frac{3.4}{-3.4} \right| = 1 \).Both values satisfy the original equation.

Key Concepts

Understanding Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. For example, the absolute value of both -3 and 3 is 3, written as \(|-3| = 3\) and \(|3| = 3\). When working with absolute value equations like \(\big| \frac{2x+3}{3x-4} \big| = 1\), it means the expression inside the absolute value can equal either 1 or -1. This is because the absolute value of both 1 and -1 is 1. To solve, we need to set up and solve two separate equations corresponding to these two possibilities.

Solving Linear Equations

Linear equations are equations of the first degree, meaning they involve terms up to the power of one, typically represented as \(ax + b = 0\). To solve the absolute value equation given in the problem \(\big|\frac{2x+3}{3x-4}\big| = 1\), we break it down into two linear equations: \(\frac{2x+3}{3x-4} = 1\) and \(\frac{2x+3}{3x-4} = -1\). Each equation is solved step-by-step:

• Multiply both sides by the denominator to clear the fraction.
• Rearrange terms to isolate the variable x.
• Solve for x to find the solutions.

For example, for \(\frac{2x+3}{3x-4} = 1\):

• Multiply both sides by \(3x-4\) to get \(2x+3 = 3x-4\).
• Rearrange to isolate x: \(x = 7\).

For \(\frac{2x+3}{3x-4} = -1\):

• Multiply both sides by \(3x-4\) to get \(2x+3 = -3x+4\).
• Rearrange to isolate x: \(x = \frac{1}{5}\) or \(x = 0.2\).

Verification of Solutions

Verification is an important step in solving equations to ensure the solutions are correct. To verify a solution, substitute the values back into the original equation and check if both sides are equal. For our solutions \(x = 7\) and \(x = 0.2\), we should check if substituting them into the original absolute value equation \(\big|\frac{2x+3}{3x-4}\big| = 1\) holds true. Let's verify:

• For \(x = 7\): \(\big|\frac{2(7) + 3}{3(7) - 4}\big| = \big|\frac{17}{17}\big| = 1\), which holds true.
• For \(x = 0.2\): \(\big|\frac{2(0.2) + 3}{3(0.2) - 4}\big| = \big|\frac{3.4}{-3.4}\big| = 1\), which also holds true.

This verification confirms that \(x = 7\) and \(x = 0.2\) are indeed solutions to the original equation.