Problem 16 - Precalculus Student Solutions Manual 5th

Question

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

Step-by-Step Solution

Answer

The solutions are \( x = \frac{7}{8} \) and \( x = \frac{1}{8} \).

Step 1: Set up the equations

Since the equation involves an absolute value, it splits into two separate equations: \[\frac{3}{2x-1} = 4 \] and \[\frac{3}{2x-1} = -4. \]

Step 2: Solve the first equation

Solve \(\frac{3}{2x-1} = 4\). Multiply both sides of the equation by \(2x-1\):\[3 = 4(2x-1)\]

Step 3: Simplify the first equation

Distribute the 4 on the right side of the equation:\[3 = 8x - 4\]Add 4 to both sides:\[3 + 4 = 8x\]So,\[7 = 8x\]Finally, divide both sides by 8:\[x = \frac{7}{8}\]

Step 4: Solve the second equation

Solve \(\frac{3}{2x-1} = -4\). Multiply both sides of the equation by \(2x-1\):\[3 = -4(2x-1)\]

Step 5: Simplify the second equation

Distribute the -4 on the right side of the equation:\[3 = -8x + 4\]Subtract 4 from both sides:\[3 - 4 = -8x\]Thus,\[-1 = -8x\]Finally, divide both sides by -8:\[x = \frac{1}{8}\]

Step 6: Present the solutions

The solutions to the equation \(\left|\frac{3}{2x-1}\right|=4\) are:\[x = \frac{7}{8} \text{ and } x = \frac{1}{8}\]

Key Concepts

solving absolute value equations

To solve absolute value equations, you need to understand what absolute value means. Absolute value represents the distance a number is from zero, regardless of direction. This means \(|a| = b\) translates to two possible equations: \(a = b\) or \(a = -b\). For example, solving \(\frac{3}{2x-1} = 4\) and \(\frac{3}{2x-1} = -4\) provides two equations that must be solved independently. Simplify the resulting expressions step by step to find the solutions.

rational expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. For example, in the given problem, \(\frac{3}{2x-1}\), 3 is the numerator and \(2x-1\) is the denominator. When solving rational expressions, you often cross-multiply to clear the fraction, simplifying the equation into a more standard linear form. Remember to check for any excluded values that might make the denominator zero, as these are not valid solutions.

precaculus

Precalculus involves mathematical skills needed to prepare for calculus, including algebraic manipulation and understanding functions. When dealing with absolute value equations involving rational expressions—as shown in the given problem—you need to use your algebra skills to isolate the variable. This step-by-step practice strengthens your ability to handle more complex equations in calculus. Always ensure you follow logical steps, such as handling the absolute value properly, splitting into two cases, and then solving each case separately.