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Understanding how to solve inequalities is a crucial part of algebra. An inequality like the one in our problem, \( 2x - 5 > 3 \), allows us to find a range of values that satisfy the given condition instead of just one solution. Here's how you can go about solving this.- **Identify the Inequality**: Look at what the inequality is asking. In our case, it's about finding all \( x \) values that make \( 2x - 5 \) greater than 3.- **Isolate the Variable**: Just like you would in a simple equation, your goal is to get \( x \) alone on one side. Start by adding or subtracting terms and then multiplying or dividing as needed. For our example:- Add 5 to both sides of the inequality to undo the \(-5\): \[ 2x - 5 + 5 > 3 + 5 \] Simplifies to: \[ 2x > 8 \] - Divide both sides by 2 to completely solve for \( x \): \[ \frac{2x}{2} > \frac{8}{2} \] Simplifies to: \[ x > 4 \]By following these steps, you've successfully isolated \( x \) and found the solution to the inequality.

Once you've solved an inequality, expressing the solution in interval notation provides a concise way to describe the range of solutions. Let's break down what this means.- **Understanding Interval Notation**: It uses parentheses and brackets to show which parts of the number line are included in the solution set. For our problem: - The parenthesis \((\) indicates that the number is not included in the solution set. - The infinity symbol \(\infty\) is always accompanied by a parenthesis, as infinity is not a number you can actually reach or include. - **Application for Our Solution**: The inequality \( x > 4 \) means that \( x \) can be any number greater than 4, but not 4 itself. Thus, in interval notation, this solution would be written as: \[ (4, \infty) \]This notation tells us that the values start just past 4 and extend infinitely to the right.

Visualizing the solution of an inequality on a number line helps understand it better. Here's how you can graph inequalities like \( x > 4 \).- **Setting Up the Number Line**: Draw a horizontal line and mark evenly spaced numbers along it.- **Plot the Critical Point**: Locate the number 4 on the line. Since \( x > 4 \), 4 itself is not part of the solution. - **Open Circle vs. Closed Circle**: - Use an open circle to denote that 4 is not included. If the inequality was \( \geq (greater \text{ or equal to}) \), you'd use a closed circle instead. - **Drawing the Arrow**: Extend an arrow from the open circle at 4 going to the right. This arrow represents all numbers greater than 4 are part of the solution. By graphing this on a number line, you can clearly see that the solution consists of all real numbers greater than 4, emphasizing that the critical point itself is not included.