Problem 32 - College Algebra Essentials

Question

Use interval notation to express solution sets and graph each solution set on a number line. Solve each linear inequality. $$-5 x \leq 30$$

Step-by-Step Solution

Answer

The solution to the inequality is \(x \geq -6\), which can be expressed in interval notation as \([-6, +\infty)\). A graph of this solution on a number line would show a closed circle on -6 and a line extending to the right.

Step 1: Solve the inequality

Starting with the inequality \(-5x \leq 30\), divide both sides by -5. Remember, since we are dividing by a negative number, the inequality flips to become \(x \geq -6\). That is the solution to the given inequality.

Step 2: Express the solution in interval notation

We know that \(x\) must be greater than or equal to -6. In interval notation, this is expressed as \([-6, +\infty)\). This means that the solution includes all numbers greater than or equal to -6.

Step 3: Graph the solution

To graph the solution on a number line, plot a closed circle on -6 (because -6 is included in the solution) and draw a line extending to the right (indicating that the solution includes all numbers greater than -6).

Key Concepts

Interval Notation

Interval notation is a simple way to represent a range of values in mathematics. When we solve an inequality, like the example \(x \geq -6\), it involves all numbers greater than or equal to -6. This is where interval notation comes in handy.

For the interval notation of our solution:

• The square bracket \([-6\) indicates that the number -6 is included in the solution set."
• The infinity symbol \(+\infty)\) means the set continues indefinitely in the positive direction."

We always use a parenthesis \()\) with infinity, as infinity is not a specific number we can include."

So, the interval notation \([-6, +\infty)\) represents all numbers from -6 to positive infinity."

Number Line Graphing

Graphing solutions on a number line gives a visual representation of the values that satisfy an inequality. For our inequality solution, \(x \geq -6\), we start with a number line and follow these steps:

• Mark the point -6 on the number line and draw a closed circle at this point. The closed circle indicates -6 is included in the solution."
• Draw a line extending to the right from -6, signifying that all numbers greater than -6 are part of the solution set."

This approach visually displays which numbers are solutions, making it easier to understand and communicate."

Using both interval notation and graphing provides a complete picture of where \(x\) falls in the solution set."

Solving Inequalities

Solving inequalities involves similar steps to algebraic equations but with a few unique rules. Let's revisit the inequality in our exercise: \(-5x \leq 30\). Here's how to solve it:

• We need to isolate \(x\) on one side of the inequality. To do this, we divide both sides by -5."
• Importantly, dividing or multiplying both sides of an inequality by a negative number requires flipping the inequality sign. This transforms \(-5x \leq 30\) into \(x \geq -6\)."

Following these steps will help you solve similar problems accurately. The key takeaway is remembering to flip the inequality when dealing with negatives."

Mastering these mechanics will make solving any linear inequality straightforward."

Inequality Properties

Understanding the properties and rules of inequalities is essential to solving them correctly. When working with inequalities, particularly note these properties:

• **Transitive Property**: If \(a > b\) and \(b > c\), then \(a > c\)."
• **Addition/Subtraction**: You can add or subtract the same number on both sides without altering the inequality."
• **Multiplication/Division by Positives**: Multiplying or dividing both sides by a positive number keeps the inequality direction the same."
• **Multiplication/Division by Negatives**: Multiplying or dividing both sides by a negative number flips the inequality sign, as seen in our exercise."

Grasping these properties allows you to handle more complex inequalities with confidence and precision. They form the backbone of solving any inequality, ensuring you're always working logically."

By keeping these properties in mind, you’ll be able to tackle each problem methodically."