Problem 25 - Algebra A Combined Function

Question

Solve each inequality. Graph the solution set. $$ -x>0 $$

Step-by-Step Solution

Answer

The solution is \(x < 0\). Graph with an open circle at 0 and shade left.

Step 1: Rewrite the Inequality

The given inequality is \(-x > 0\). To isolate \(x\), we need to multiply both sides of the inequality by \(-1\). Remember that multiplying or dividing an inequality by a negative number reverses the inequality sign. This changes the inequality to \(x < 0\).

Step 2: Determine the Solution Set

With the inequality \(x < 0\), the solution set consists of all numbers that are less than zero. This includes all negative numbers and does not include zero itself.

Step 3: Graph the Solution Set

To graph the solution set \(x < 0\), draw a number line. Put an open circle at 0 to denote that 0 is not included, and shade the entire line to the left of 0, indicating all numbers less than 0 are included in the solution set.

Key Concepts

Inequality Manipulation

Manipulating inequalities involves changing the form of an inequality to isolate the variable of interest, often using similar operations as in equations. However, there's a crucial difference: if you multiply or divide both sides of an inequality by a negative number, the inequality sign must be reversed.
In the inequality \(-x > 0\), the goal is to isolate \(x\). To do this, we multiply both sides by \(-1\). The essential rule to remember here is to flip the inequality sign when performing this operation, converting \(>\) into \(<\). This transforms the inequality into \(x < 0\).
Other operations such as adding or subtracting a number from both sides do not require changing the inequality sign. However, always check your steps to ensure accuracy when solving inequalities.

Graphing Solutions

Graphing solutions for inequalities helps visualize the range of values that satisfy the inequality. Let's look at the inequality \(x < 0\).
To graph this, start by drawing a number line. This line represents all potential values of \(x\). Locate zero on the line as a reference point. Since the inequality specifies \(< 0\), you place an open circle at zero to indicate that zero itself is not included.
The next step is shading the line. For \(x < 0\), you shade the entire portion of the line to the left of zero. This shaded area visually represents all numbers less than zero, indicating the solution set for the inequality.

Properties of Inequalities

Understanding the properties of inequalities is fundamental in solving and graphing them correctly. Here are some key properties you should know:

• Addition/Subtraction Property: Adding or subtracting the same number on both sides maintains the inequality.
• Multiplication/Division with Positive Numbers: Multiplying or dividing both sides by a positive number does not change the inequality's direction.
• Multiplication/Division with Negative Numbers: This is crucial - it requires reversing the inequality sign to maintain the correct relationship between the terms.
• Transitive Property: If \(a > b\) and \(b > c\), then \(a > c\).

By leveraging these properties, you can solve inequalities more effectively and ensure the solutions accurately reflect the problem's constraints. Applying these consistently helps avoid errors, especially when manipulating and simplifying complex inequalities.