Problem 28 - College Algebra and Calculus: An Applied Approach

Question

Solve the inequality. Then graph the solution set on the real number line. \(\frac{1}{x}<4\)

Step-by-Step Solution

Answer

The solution to the inequality \(\frac{1}{x}<4\) is \(x > \frac{1}{4}\). On a number line, this is represented by a parenthesis on \(\frac{1}{4}\) and an arrow pointing to the right towards \(\infty\).

Step 1: Clear the fraction

In order to remove the fraction from the inequality, multiply both sides by x, keeping in mind that the direction of inequality changes if x is negative. For x > 0, we get \(x * (\frac{1}{x}) < 4x\), simplifying to \(1 < 4x\). However, if x < 0, the inequality is reversed, giving \(1 > 4x\).

Step 2: Solve the inequality

Dividing both sides by 4 in each inequality yields \(x > \frac{1}{4}\) when x > 0 and \(x < \frac{1}{4}\) when x < 0. The inequality \(x < \frac{1}{4}\) for \(x < 0\) doesn't contribute any new solutions, so it can be ignored. The final solution is \(x > \frac{1}{4}\).

Step 3: Graph the solution

the solution \(x > \frac{1}{4}\) is represented on a number line by a parenthesis on \(\frac{1}{4}\), signifying that \(\frac{1}{4}\) itself is not included in the solutions, and an arrow pointing towards the right to \(\infty\), indicating all numbers greater than \(\frac{1}{4}\) are solutions.

Key Concepts

Real Number Line

The real number line is a visual representation of real numbers arranged in increasing order from left to right. Think of it as an infinite line extending in both directions, where each point corresponds to a unique real number.
In this exercise, we use the real number line to demonstrate the set of solutions for the inequality we have solved. The real number line helps us visualize these solutions effectively.Here are some key components when working with the real number line:

• Points: Specific numbers or solutions are shown as points on the line.
• Intervals: A range of numbers that satisfy an inequality is depicted as a portion of the line.
• Direction: Arrows signify continuity of numbers in a positive, negative, or both directions.

In our solution, we denote the interval starting just after \(rac{1}{4}\), extending towards positive infinity, showing all values of \( x\) that meet our condition.

Inequality Solving

Solving inequalities involves finding the set of numbers that make the inequality true. This can be similar to solving equations, but with a few extra considerations.When working with inequalities, remember:

• Direction: The inequality sign ("><", "≤", ">=", ">") indicates the relationship between the two sides.
• Reverse Sign: Multiplying or dividing by a negative value reverses the inequality sign.
• Simplify: Just like equations, simplify both sides to isolate the variable.

In our example, \(\frac{1}{x}<4\), we made sure to treat cases where \(x\) could change the inequality's direction, particularly when \(x < 0\). Always check these conditions to ensure a correct solution.

Fractional Inequalities

Fractional inequalities are inequalities that contain fractions. Solving them often requires special techniques to handle the fractions properly.To solve \(\frac{1}{x}<4\), we had to clear the fraction. We multiplied through by \(x\), remembering that:

• Same Sign: Multiplying by a positive maintains the inequality direction.
• Reverse Sign: Multiplying by a negative reverses it, especially relevant if \(x\) may be negative.

After clearing fractions, solve the resulting inequalities individually. In this case, note how carefully analyzing sign changes allowed accurate problem solving.

Graphing Solutions

Graphing the solution of inequality provides a visual representation to quickly understand which numbers satisfy the condition.When graphing \(x > \frac{1}{4}\):

• Open Circle or Parenthesis: Use these to indicate that the value \(\frac{1}{4}\) itself is not included as a solution.
• Arrow: Draw an arrow extending to the right to show all numbers greater than \(\frac{1}{4}\) are included.

By placing the parenthesis at \(\frac{1}{4}\) and drawing the arrow towards the right, you clearly depict the solution set. The real number line then reflects this understanding visually, making it easier to identify and verify solutions.