Question

Express the solution set of the given inequality in interval notation and sketch its graph. $$ \frac{2}{x}<5 $$

Short answer

The solution set is \(\left( \frac{2}{5}, \infty \right)\).

Step-by-step solution

Clear the Fraction by Multiplication

Start by eliminating the fraction. Multiply both sides of the inequality by the variable denominator, which is \(x\). This gives us \(2 < 5x\).

Solve the Inequality for x

Divide both sides by 5 to solve for \(x\). This results in \(x > \frac{2}{5}\). Make sure to keep the inequality direction the same since we are dividing by a positive number.

Consider the Domain of the Original Inequality

The original inequality \(\frac{2}{x} < 5\) implies that \(x\) cannot be zero because division by zero is undefined. This seeks to ensure that \(x eq 0\). However, in this context, \(x>\frac{2}{5}\), which automatically satisfies \(xeq 0\).

Express the Solution Set in Interval Notation

The solution \(x > \frac{2}{5}\) is represented in interval notation as \(\left( \frac{2}{5}, \infty \right)\).

Graph the Solution on a Number Line

To graph the inequality, draw a number line, and place an open circle on \(\frac{2}{5}\) to show that it is not included in the solution. Then, shade the line to the right of \(\frac{2}{5}\) to indicate that all numbers greater than \(\frac{2}{5}\) are part of the solution.

Key concepts

Understanding Inequalities

Inequalities are like equations, but instead of equating two expressions, they compare them. Whenever you read an inequality, it shows how one side stands in relation to the other. For instance, in our example \( \frac{2}{x} < 5 \), it tells us that the expression \( \frac{2}{x} \) is less than 5. It's like saying one number is 'lower than', 'greater than', or 'different from' another.
Keep in mind these common inequality symbols:

• \(<\) means 'less than'
• \(>\) means 'greater than'
• \( \leq \) means 'less than or equal to'
• \( \geq \) means 'greater than or equal to'

Dealing with inequalities often means looking for the range of numbers that make an inequality true, leading us to the concept of a 'solution set' and how to express it in 'interval notation.'

Exploring the Solution Set

The solution set of an inequality includes all values that make the inequality true. In the inequality \( \frac{2}{x} < 5 \), we were able to manipulate it to \( x > \frac{2}{5} \). This means any number greater than \( \frac{2}{5} \) will satisfy the inequality.
Expressing the solution set in interval notation provides a convenient way to write out this range of values. Interval notation uses brackets to show which numbers are included in the set. Here’s how it works:

• Parentheses or \((\) are used when a number is not included. It's an 'open' interval, meaning just slightly after that number, the values count.
• Brackets or \([\) are used when a number is included. It's a 'closed' interval, which means starting right at that number.

For our solution, \( x > \frac{2}{5} \) translates to \( \left( \frac{2}{5}, \infty \right) \) in interval notation, highlighting all numbers greater than but not including \( \frac{2}{5} \). The infinity symbol \( \infty \) suggests the numbers continue on to infinitely large, or indefinitely greater values.

Graphing the Inequality

Graphing is a visual way to represent the solution of an inequality on a number line. It helps you see the range of possible solutions at a glance. For our inequality \( x > \frac{2}{5} \), the steps to graph it are straightforward:
Start by drawing a horizontal line to represent the number line. Mark the point \( \frac{2}{5} \) on this line. Since \( \frac{2}{5} \) is not part of the solution (the inequality is 'greater than' but not 'greater than or equal to'), we use an open circle on \( \frac{2}{5} \) to demonstrate this.
Next, shade or draw an arrow to the right of \( \frac{2}{5} \). This shows that all numbers to the right, which are larger, are included in the solution set. By visualizing it this way, it's easy to confirm our interval notation \( \left( \frac{2}{5}, \infty \right) \). It communicates clearly that the solution involves every number greater than \( \frac{2}{5} \) stretching towards infinity.