Problem 91 - A Graphical Approach to College Algebra

Question

Solve each inequality analytically. Write the solution set in interval notation. Support your answer graphically. $$\frac{1}{3} x-\frac{1}{5} x \leq 2$$

Step-by-Step Solution

Answer

The solution set is \((-\infty, 15]\) in interval notation.

Step 1: Combine Like Terms

Start by simplifying the left side of the inequality by combining like terms, which are the terms involving the variable \(x\). The expression \( \frac{1}{3}x - \frac{1}{5}x \) can be rewritten using a common denominator of 15. So, we have: \[ \frac{1}{3} x - \frac{1}{5} x = \frac{5}{15} x - \frac{3}{15} x = \frac{2}{15} x \]. This gives us the inequality \( \frac{2}{15} x \leq 2 \).

Step 2: Solve for x

To isolate \(x\), multiply both sides of the inequality by the reciprocal of \( \frac{2}{15} \), which is \( \frac{15}{2} \). Remember that multiplying or dividing both sides of an inequality by a positive number keeps the inequality sign the same: \[ x \leq 2 \times \frac{15}{2} \]. Simplifying the right side, we find: \( x \leq 15 \).

Step 3: Write the Solution in Interval Notation

The solution \( x \leq 15 \) can be expressed in interval notation as \(( -\infty, 15 ]\). This notation indicates that \( x \) can take any value less than or equal to 15.

Step 4: Graph the Solution

To graph this solution on a number line, draw a solid circle (or include a filled-in dot) at \(x = 15\) to show that it is included in the solution set. Shade the region to the left of 15 to represent all numbers less than 15. This visually shows the inequality \(x \leq 15\).

Key Concepts

Interval Notation

Interval notation is a concise way of expressing ranges of values, often used with inequalities to clearly define a set of numbers. For the inequality \( x \leq 15 \), we use interval notation to express all possible values of \( x \). This format provides a straightforward representation:

• Parentheses \(( )\) indicate numbers that are not included in the range,
• Brackets \([ ]\) suggest numbers that are included.

Thus, for \( x \leq 15 \), the interval notation is \((-\infty, 15 ]\). This means the solution includes all numbers less than or equal to 15, extending infinitely in the negative direction. The number 15 is enclosed in a bracket to show it is included in the solution.
By using interval notation, we simplify communication of complex numerical ranges in a format that's widely understood in mathematics.

Graphical Solution

Graphical solutions involve representing inequalities on a number line, providing a visual understanding of the solution set. For the inequality \(x \leq 15\), the graphical depiction helps illuminate the range of possible solutions:

• Begin by identifying the point \(x = 15\) on the number line.
• Use a solid circle at this point to indicate that 15 is included in the solution set. A solid circle represents inclusion, while an open circle would show exclusion.
• Shade the line extending leftward from 15 to represent all values that are less than 15.

This shaded portion visually confirms that any number less than or equal to 15 satisfies the inequality. The graphical solution provides an intuitive understanding of how inequalities function, reinforcing the solution calculated analytically.

Combining Like Terms

Combining like terms is a fundamental step in simplifying expressions, particularly in algebraic inequalities and equations. It involves merging terms that have the same variable and exponent, making the expression simpler. Consider the given inequality \( \frac{1}{3}x - \frac{1}{5}x \leq 2 \):

• Identify terms that are similar—in this context, both terms involve \(x\).
• Convert them to a common denominator for easy subtraction, which in this case is 15.
• Rewrite \(\frac{1}{3}x\) as \(\frac{5}{15}x\) and \(\frac{1}{5}x\) as \(\frac{3}{15}x\).
• Subtract these like terms: \(\frac{5}{15}x - \frac{3}{15}x = \frac{2}{15}x\).

This simplification streamlines the inequality, making it easier to solve for \(x\). By combining like terms, you reduce complexity, avoid possible calculation errors, and set the stage for further manipulations, such as isolating the variable.