Problem 24 - Algebra A Combined Function

Question

Solve each inequality. Graph the solution set. $$ -5 x<20 $$

Step-by-Step Solution

Answer

The solution is \(x > -4\) and it is graphed as an open circle at \(-4\) extending right.

Step 1: Isolate the Variable

To solve the inequality \(-5x < 20\), we need to isolate \(x\) on one side. We can do this by dividing each side of the inequality by \(-5\). Remember, when dividing an inequality by a negative number, we must flip the inequality sign. Divide both sides to get: \(x > -4\).

Step 2: Graph the Solution Set

On a number line, represent the solution \(x > -4\). Place an open circle at \(-4\) to indicate that \(-4\) is not included in the solution. Draw a ray extending to the right from \(-4\) to represent all numbers greater than \(-4\).

Key Concepts

Isolating Variables

Isolating the variable is a crucial step when solving inequalities because it helps to find what values a variable can take. Let's take a closer look at how to do this efficiently.To isolate a variable like in the inequality \(-5x < 20\), our main goal is to have the variable \(x\) on one side by itself. Here’s how you can achieve it:

• First, consider the operation currently performed on \(x\), which in this case is multiplication by \(-5\).
• To undo this, perform the opposite operation. Divide both sides of the inequality by \(-5\).
• It's important to remember a critical rule: Whenever you divide or multiply both sides of an inequality by a negative number, you must flip the inequality sign. This means \(<\) becomes \(>\).

Executing these steps, we have \(x > -4\).By isolating \(x\), you get a clearer picture of the range \(x\) belongs to and prepare yourself for graphing the solutions.

Graphing Solutions

Graphing the solutions of an inequality gives a visual representation of all the possible values the variable can take. Let’s learn how to graph the solution \(x > -4\) on a number line effectively.

• Start by drawing a straight horizontal line, which will serve as your number line.
• Identify the critical point from the solution, which is \(-4\) in this scenario.
• Because the inequality is a strict \(>\) (greater than but not equal to), indicate this by placing an open circle on the number line at \(-4\). The open circle shows that \(-4\) is not included in the solution.
• Next, draw a ray (a line with an arrow) starting at \(-4\) and extending to the right. This represents all numbers greater than \(-4\).

This simple graphical method helps clearly indicate the solution set on a number line, making it easier to interpret the inequality's results.

Inequality Rules

Understanding inequality rules is vital when solving them, as they guide how operations affect the inequality sign and ultimately the solution.Here are some essential rules to always remember:

• **Transposing terms and maintaining the normal order:** If you add or subtract a term from both sides, the direction of the inequality remains unchanged.
• **Multiplying or dividing by a positive number:** You can multiply or divide both sides of an inequality by a positive number without changing the inequality sign.
• **Multiplying or dividing by a negative number:** When you multiply or divide both sides by a negative number, you must switch the direction of the inequality sign (\(<\) becomes \(>\) and vice versa). This is crucial as not doing so would result in an incorrect solution.
• **Combining inequalities:** When dealing with compound inequalities (e.g., \(a < x < b\)), ensure operations done on all parts are consistent to maintain truth.

By understanding and applying these fundamental rules, you'll be more adept at solving inequalities correctly and efficiently.