Problem 35 - Algebra A Combined Function

Question

Solve each inequality. Write each answer using solution set notation. $$ 4 x>1 $$

Step-by-Step Solution

Answer

\( \{ x \mid x > \frac{1}{4} \} \)

Step 1: Understand the Inequality

We start with the inequality \( 4x > 1 \). This means we need to find all values of \( x \) that make this statement true.

Step 2: Isolate the Variable (x)

To isolate \( x \), divide both sides of the inequality by 4. This gives:

Step 2.1: Division

\[ x > \frac{1}{4} \] This simplifies the inequality, leaving \( x \) by itself on one side.

Step 3: Write the Solution in Interval Notation

The inequality \( x > \frac{1}{4} \) represents all numbers greater than \( \frac{1}{4} \). In interval notation, this is expressed as \( \left( \frac{1}{4}, \infty \right) \).

Step 5: Conclusion: Solution Set Notation

Thus, the solution set in notation form is \( \{ x \mid x > \frac{1}{4} \} \). This reads as "the set of all \( x \) such that \( x \) is greater than \( \frac{1}{4} \)."

Key Concepts

Solution Set Notation

Solution set notation is a way to express the set of all solutions to an inequality. When you solve an inequality like \( 4x > 1 \), you find that \( x > \frac{1}{4} \). To express this in solution set notation, you would write it as

• \( \{ x \mid x > \frac{1}{4} \} \)

This reads as "the set of all \( x \) such that \( x \) is greater than \( \frac{1}{4} \)."
It's like saying, "Here are all the possible values of \( x \) that make the inequality true." Solution set notation is very precise and is commonly used in mathematical writing.
It's essential to understand this format, as it quickly tells you all the possible solutions without listing every single one.

Interval Notation

Interval notation offers another way to express the solution to an inequality. It shows a range of numbers between specific endpoints.
For the inequality \( x > \frac{1}{4} \), the range of \( x \) is all numbers greater than \( \frac{1}{4} \). Here's how we write that in interval notation:

• \( \left( \frac{1}{4}, \infty \right) \)

The round bracket "\( ( \)" next to \( \frac{1}{4} \) indicates that \( \frac{1}{4} \) is not included in the set of solutions (just greater than \( \frac{1}{4} \)).
The infinity symbol "\( \infty \)" on the far right shows that there is no upper limit to the values of \( x \).
Interval notation is useful because it provides a concise and efficient way to describe an endless range of numbers. It’s often used in calculus and higher-level math courses.

Isolate the Variable

To solve any inequality successfully, it's crucial to isolate the variable. This means getting the variable on one side of the inequality sign, all by itself. In the inequality \( 4x > 1 \), you want to get \( x \) alone.
Here's how you do it:

• First, you need to perform the same operation on both sides of the inequality. To 'undo' the multiplication of \( 4 \), divide both sides by \( 4 \).
• This gives you \( x > \frac{1}{4} \)).

Now, \( x \) is isolated, and the inequality tells you clearly all the values for \( x \).
Isolating the variable is a fundamental skill in algebra that helps you find the solution to equations and inequalities. It sets the groundwork for more complex problem solving and mathematical manipulations.