Problem 70 - A Graphical Approach to College Algebra

Question

An equation of the form \(|f(x)|=|g(x)|\) is given. (a) Solve the equation analytically and support the solution graphically. (b) Solve \(|f(x)|>|g(x)|\) (c) Solve \(|f(x)|<|g(x)|\) $$|-5 x+1|=|3 x-4|$$

Step-by-Step Solution

Answer

Solutions: (a) \(x=\frac{5}{8}\), \(x=-\frac{3}{2}\); (b) \(-\frac{3}{2} < x < \frac{5}{8}\); (c) \(x > \frac{5}{8}\) or \(x < -\frac{3}{2}\).

Step 1: Analyze the equation

The equation given is \(|-5x + 1| = |3x - 4|\). This equation can be solved by considering different cases based on when each expression inside the absolute value is positive or negative.

Step 2: Set up the cases for solving

We must consider two main cases to solve the equation: 1. \(-5x + 1 = 3x - 4\)2. \(-5x + 1 = -(3x - 4)\)These cases arise from the property that \(|a| = |b|\) implies either \(a = b\) or \(a = -b\).

Step 3: Solve the first case

The first case is \(-5x + 1 = 3x - 4\). Solving for \(x\), we get:\[-5x + 1 = 3x - 4\]Add \(5x\) to both sides:\[1 = 8x - 4\]Add 4 to both sides:\[5 = 8x\]Divide by 8:\[x = \frac{5}{8}\]

Step 4: Solve the second case

The second case is \(-5x + 1 = -(3x - 4)\). Solving for \(x\), we get:\[-5x + 1 = -3x + 4\]Add \(5x\) to both sides:\[1 = 2x + 4\]Subtract 4 from both sides:\[-3 = 2x\]Divide by 2:\[x = -\frac{3}{2}\]

Step 5: Solve \(|f(x)| > |g(x)|\)

We analyze the inequality \(|-5x + 1| > |3x - 4|\) by considering two separate inequalities:1. \(-5x + 1 > 3x - 4\)Solving, we get:\[1 + 4 > 3x + 5x\]\[5 > 8x\]\[x < \frac{5}{8}\]AND2.\(-5x + 1 < -(3x - 4)\)Solving, we get:\[1 < -3x + 4 + 5x\]\[1 < 2x + 4\]\[-3 < 2x\]\[x > -\frac{3}{2}\]Thus, the solution is the interval \(-\frac{3}{2} < x < \frac{5}{8}\).

Step 6: Solve \(|f(x)| < |g(x)|\)

Using the same approach, analyze the inequality \(|-5x + 1| < |3x - 4|\):1.\(-5x + 1 < 3x - 4\)Solving, we get:\[1 + 4 < 3x + 5x\]\[5 < 8x\]\[x > \frac{5}{8}\]OR2.\(-5x + 1 > -(3x - 4)\)Solving, we get:\[1 > -3x + 4 + 5x\]\[1 > 2x + 4\]\[-3 > 2x\]\[x < -\frac{3}{2}\]The solution is \(x > \frac{5}{8}\) or \(x < -\frac{3}{2}\).

Key Concepts

Inequalities

Understanding inequalities in the context of absolute value equations is crucial. Inequalities tell us about the size or order of the expression outcomes. Here, we are dealing with absolute values, which are always non-negative. To solve inequalities involving absolute values, such as \(|f(x)| > |g(x)|\) and \(|f(x)| < |g(x)|\), we must consider different scenarios where one expression is larger or smaller than the other.

- For \(|f(x)| > |g(x)|\), we have two conditions to satisfy: \(-5x + 1 > 3x - 4\) and \(-5x + 1 < -(3x - 4)\). This creates a situation where we identify an interval, here \(-\frac{3}{2} < x < \frac{5}{8}\), as the solution where this inequality holds true.
- For \(|f(x)| < |g(x)|\), the conditions \(-5x + 1 < 3x - 4\) and \(-5x + 1 > -(3x - 4)\) need to be fulfilled, giving us solutions outside the interval found in the first inequality, \(x > \frac{5}{8}\) or \(x < -\frac{3}{2}\).

These solutions show where the respective expressions inside the absolute values dominate, solving these expressions is about splitting into these cases to examine each possible outcome.

Graphical Representation

Visualizing absolute value equations can simplify understanding of inequalities and their solutions. By graphing the expressions \(-5x + 1\) and \(3x - 4\), you can see the points where their absolute values are equal or one exceeds the other. Consider these steps:

- **Overlay graphs**: Plot both \(-5x + 1\) and \(3x - 4\) on a coordinate plane. Notice where the lines intersect. These intersection points sometimes correspond to solutions of the equation \(|-5x + 1| = |3x - 4|\).
- **Understanding regions**: Identify regions above and below these graphs to see where inequalities hold true.
- **Intersection points**: Check specific x-values where the graphs intersect with the x-axis. These solutions confirm the intervals derived algebraically.

Graphs make relationships between functions visible effectively and assist in confirming analytical solutions through visual means.

Algebraic Solutions

Solving absolute value equations like \(|-5x + 1| = |3x - 4|\) involves understanding how absolute values react under different conditions. This requires breaking the problem into cases to remove absolute value bars.

- **Case 1**: Assume \(-5x + 1 = 3x - 4\). Solving yields \(x = \frac{5}{8}\). This handles when values inside both absolute expressions are equal.
- **Case 2**: Consider \(-5x + 1 = -(3x - 4)\). Solving results in \(x = -\frac{3}{2}\). This case refers to when one expression is the negative of the other.

For inequalities, we treat both possible scenarios separately: one expression being greater or lesser than another. Solving these individually provides insight into the range of x-values that make the inequality true. Algebra relies heavily on recognizing and exploiting these scenarios to achieve solutions.