Problem 47 - Intermediate Algebra

Question

Solve each inequality. Graph the solution set and write it using interval notation. $$ \frac{2}{5}>\frac{4}{5} x $$

Step-by-Step Solution

Answer

The solution is \( x < \frac{1}{2} \) or \((-\infty, \frac{1}{2})\).

Step 1: Isolate the Variable

Start by isolating the variable on one side of the inequality. We have the inequality \( \frac{2}{5} > \frac{4}{5}x \). To isolate \( x \), first multiply both sides by \( 5 \) to eliminate the fractions: \[ 2 > 4x \].

Step 2: Solve the Inequality

Next, solve for \( x \) by dividing both sides of the inequality by \( 4 \):\[ x < \frac{2}{4} \].Simplify \( \frac{2}{4} \) to get:\[ x < \frac{1}{2} \].

Step 3: Graph the Solution Set

To graph the solution \( x < \frac{1}{2} \), draw a number line. Mark the number \( \frac{1}{2} \) on the line with an open circle, indicating that \( \frac{1}{2} \) is not included in the solution. Shade the region to the left of \( \frac{1}{2} \) to signify that all numbers less than \( \frac{1}{2} \) are part of the solution set.

Step 4: Write the Solution in Interval Notation

In interval notation, the solution \( x < \frac{1}{2} \) is written as:\( (-\infty, \frac{1}{2}) \). This represents all values starting from negative infinity up to, but not including, \( \frac{1}{2} \).

Key Concepts

Interval Notation

Interval notation is a way of writing subsets of the real number line. It provides a concise method to express solutions for inequalities. Think of it as a shorthand for describing a range of possible values that satisfy a condition.

For example, the interval

• \((-\infty, \frac{1}{2})\): This notation represents all real numbers that are less than \(\frac{1}{2}\). The round bracket '(' means that the endpoint, \(\frac{1}{2}\), is not included in the interval.
• \((a, b)\): Here, both endpoints \(a\) and \(b\) are not included in the interval.
• \([a, b]\): When square brackets are used, both \(a\) and \(b\) are included in the solution set.
• \((a, b]\): In this mix, \(a\) is not included, but \(b\) is included.

Remember, open intervals use parentheses, indicating that the endpoints are not part of the solution. Closed intervals use brackets to show they are part of the solution.

This notation helps in cataloging all values that satisfy an inequality without listing them individually.

Number Line Graphing

Graphing solutions on a number line offers a visual representation of an inequality's solution set. It helps understand which numbers are included or excluded in the solution.

To graph the inequality \(x < \frac{1}{2}\), follow these steps:

• Draw a horizontal line, this will be your number line.
• Identify and mark the number \(\frac{1}{2}\) on the line.
• Since the inequality is less than, and not less than or equal to, we use an open circle at \(\frac{1}{2}\). This denotes that \(\frac{1}{2}\) is not included in the solution.
• Shade or draw an arrow to the left of \(\frac{1}{2}\), indicating all numbers less than \(\frac{1}{2}\) are solutions.

Number line graphing is especially helpful when visualizing concepts such as unions or intersections of multiple intervals. It aids in confirming that no number is left out from the range you're interested in.

Isolate Variable

Isolating the variable is a crucial step in solving inequalities and equations. It involves manipulating the expression so the variable stands alone on one side of the inequality or equation. This process helps determine the possible values that satisfy the condition.

For instance, in the inequality \(\frac{2}{5} > \frac{4}{5}x\), isolating \(x\) involves:

• Eliminating fractions by multiplying both sides by 5, simplifies to \(2 > 4x\).
• Dividing both sides by 4 to solve for \(x\), gives us \(x < \frac{1}{2}\).

By isolating \(x\), we discover which numbers satisfy the given inequality.

• Attention must be given when multiplying or dividing both sides by a negative number, as this action reverses the inequality sign.

The ultimate goal is to have the variable on one side, ensuring clarity when interpreting the inequality's solution.