Problem 47 - College Algebra with Modeling and Visualization

Question

Solve the equation (a) graphically, (b) numerically, and (c) symbolically. Then solve the nelated inequality. $$|5-3 x|=2, \quad|5-3 x|>2$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 1\) and \(x = \frac{7}{3}\). The inequality solution is \(x < 1\) or \(x > \frac{7}{3}\).

Step 1: Understanding the Equation

The equation \(|5 - 3x| = 2\) involves finding \(x\) where the absolute value of \(5 - 3x\) is equal to 2. Absolute value equations often split into two separate equations due to the nature of absolute values.

Step 2: Solving Graphically

To solve graphically, plot two functions: \(y = |5 - 3x|\) and \(y = 2\). The solutions to the equation \(|5 - 3x| = 2\) are the \(x\)-coordinates where these two graphs intersect.

Step 3: Plotting Functions

The function \(y = |5 - 3x|\) will be V-shaped, and \(y = 2\) is a horizontal line. The points of intersection can be seen by plotting these on a graph.

Step 4: Graphical Results

Upon graphing, you will observe that the two lines intersect at the points \(x = 1\) and \(x = \frac{7}{3}\). These are the graphical solutions to the equation \(|5 - 3x| = 2\).

Step 5: Solving Numerically

Verify graphically obtained values. Calculate for \(x = 1\), which gives \(|5 - 3(1)| = 2\), and for \(x = \frac{7}{3}\), check if \(|5 - 3(\frac{7}{3})| = 2\). Both satisfy the equation.

Step 6: Solving Symbolically

Break the equation into two separate equations based on the definition of absolute value: \(5 - 3x = 2\) and \(5 - 3x = -2\).

Step 7: Solving Simplified Equations

Solving \(5 - 3x = 2\): 1. Subtract 5 from both sides: \(-3x = -3\).2. Divide both sides by -3: \(x = 1\).Solving \(5 - 3x = -2\):1. Subtract 5 from both sides: \(-3x = -7\).2. Divide both sides by -3: \(x = \frac{7}{3}\).

Step 8: Inequality Analysis

Consider the inequality \(|5 - 3x| > 2\). The value of \(5 - 3x\) is greater or less than 2, giving 1. \(5 - 3x > 2\) or 2. \(5 - 3x < -2\).

Step 9: Simplifying the Inequality

Solving \(5 - 3x > 2\):1. Subtract 5 from both sides: \(-3x > -3\).2. Divide both sides by -3 (flip the inequality): \(x < 1\).Solving \(5 - 3x < -2\):1. Subtract 5 from both sides: \(-3x < -7\).2. Divide both sides by -3 (flip the inequality): \(x > \frac{7}{3}\).

Step 10: Inequality Solution

The solution to the inequality \(|5 - 3x| > 2\) is the set of \(x\) values \(x < 1\) or \(x > \frac{7}{3}\).

Key Concepts

Absolute Value

The absolute value of a number refers to its distance from zero on the number line, without considering direction. In mathematical terms, the absolute value of a number, say \( a \), is denoted as \(|a|\) and is always positive. For example, both \(|3|\) and \(|-3|\) are equal to 3, since they are both 3 units away from zero.

When dealing with absolute value in equations, it creates two potential scenarios: one where the expression inside is positive and another where it is negative. To solve an absolute value equation like \(|5 - 3x| = 2\), we consider both cases separately:

• Case 1: \(5 - 3x = 2\)
• Case 2: \(5 - 3x = -2\)

This bifurcation into two scenarios allows us to determine all possible solutions. Absolute values, therefore, make us think about multiple perspectives in a single equation.

Graphical Solutions

Graphical solutions involve visually interpreting equations by plotting them on a coordinate system. For the equation \(|5 - 3x| = 2\), graphical representation helps us "see" where solutions exist.

We can start by plotting the graph of \(y = |5 - 3x|\), which forms a V-shape due to its absolute value nature. This curve reflects its shape across the x-axis. Next, plot the line \(y = 2\), which is a horizontal line.

• The intersection points of these plots represent solutions to our original equation.
• In this particular exercise, the intersections occur at \(x = 1\) and \(x = \frac{7}{3}\).

Graphical methods give a clear visual insight that complements analytical solutions.

Numerical Solutions

Numerical solutions involve the direct calculation of values to verify solutions obtained from other methods. After finding potential solutions graphically or symbolically, plugging these back into the original equation allows us to check our work.

For example, take \(x = 1\) and \(x = \frac{7}{3}\) found graphically:

• Substitute \(x = 1\) into \(|5 - 3 \times 1|\): This simplifies to \(|2|\), which equals 2, confirming a solution.
• Next, substitute \(x = \frac{7}{3}\) into \(|5 - 3 \times \frac{7}{3}|\): Simplifies to \(|-2|\), again evaluating to 2, verifying another solution.

This step ensures that graphical intersections hold the same results as logical evaluations, cementing the found solutions as accurate.

Symbolic Solutions

Symbolic solutions use algebraic manipulations to solve equations. When solving an absolute value equation symbolically, you start by breaking down the original equation involving the absolute value into two separate linear equations.

For \(|5 - 3x| = 2\):

• Equation 1: \(5 - 3x = 2\). Solve to find \(x = 1\).
• Equation 2: \(5 - 3x = -2\). Solve to find \(x = \frac{7}{3}\).

These simplifications are straightforward algebraic steps that lead to a solution without needing a graph. This method relies on recognizing that solving absolute value involves considering both possible conditions under which the expression takes on the initial equation's value.

Inequalities

Inequalities define a range of solutions, rather than specific points. For the inequality \(|5 - 3x| > 2\), it asks for the values of \(x\) where the absolute value expression exceeds 2. Similar to equations, inequalities with absolute values also split into two scenarios:

• Case 1: Express \(5 - 3x > 2\). Simplify to find \(x < 1\).
• Case 2: Express \(5 - 3x < -2\). Simplify to find \(x > \frac{7}{3}\).

When you solve these inequalities, ensure to flip the inequality sign when dividing by a negative number.

The solutions show intervals on the number line - values that do satisfy \(|5 - 3x| > 2\) are those less than 1 or greater than \(\frac{7}{3}\). Thus, inequalities provide insight into the range of potential solutions.