Problem 66 - Introductory Algebra for College Students

Question

Use both the addition and multiplication properties of inequality to solve each inequality and graph the solution set on a number line. \(3-x \geq-3\)

Step-by-Step Solution

Answer

The solution to the inequality \(3 - x \geq -3\) is \(x \leq 6\). In the number line, this is represented by a closed dot at 6 and an arrow pointing towards the left, towards negative infinity.

Step 1: Using Additive Inverses

Add x to both sides of the inequality 3 - x ≥ -3 to cancel out the -x on the left side. This results in: 3 ≥ x - 3

Step 2: Simplify the Inequality

Simplify the inequality by adding 3 to both sides. This will isolate x on the left side. The simplified inequality is: 6 ≥ x.

Step 3: Flip the Inequality

As x is generally represented on the left side of the inequality for easier understanding, flip the inequality sign and also the sides which transposes the inequality into: x ≤ 6.

Step 4: Graphing the Solution

On a number line, represent x ≤ 6 by a closed dot at 6 (as 6 is included in the solution) with an arrow extending to the left towards negative infinity (since x can be any number less than or equal to 6).

Key Concepts

Addition Property of Inequality

When solving inequalities, the addition property of inequality helps maintain the truth of the statement. This property states that if you add the same number to both sides of an inequality, the inequality remains true. For example, if you have an inequality like 3 - x ≥ -3, you can add 'x' to both sides to get rid of the '-x' on the left. This will transform the inequality to 3 ≥ x - 3. It's crucial to keep the inequality balanced by adding the same value to each side.

• This step helps eliminate terms, making it easier to isolate the variable you're solving for.
• By maintaining balance, you ensure the solution accurately reflects the inequality.

As you continue solving, keeping this in mind contributes to clear and error-free results.

Multiplication Property of Inequality

The multiplication property of inequality is another crucial concept. It indicates that multiplying or dividing both sides of an inequality by a positive number does not change the inequality's direction. However, when multiplying or dividing by a negative number, the inequality sign must flip. In our specific problem, 6 ≥ x, we don't need to multiply or divide by negative numbers, so the inequality direction remains unchanged.
However, always remember:

• Multiplying or dividing by positive numbers keeps the inequality symbol the same.
• Multiplying or dividing by negative numbers requires flipping the inequality symbol.

This principle assures that the inequality is true and correctly represented, essential for finding the correct solution.

Graphing Inequalities

Graphing inequalities involves showing the solution set visually on a number line. To graph an inequality like x ≤ 6, we focus on representing all numbers that make the inequality true. You'll notice a closed dot is used above the number 6. This symbolizes that 6 is part of the solution (since the inequality includes 'equal to').
Here are some guidelines:

• Use a closed dot when the inequality is "less than or equal to", or "greater than or equal to" as it includes the endpoint.
• Use an open dot for "less than" or "greater than", excluding the endpoint.
• Draw an arrow from the dot extending left or right, depending on whether the inequality is less than or greater than.

By following these rules, you create a clear graphical representation of all numbers that can be solutions, making it easier to visualize and understand the solutions.

Number Line

The number line is a fundamental tool in math used to visualize numbers and operations, including inequalities. It helps you quickly grasp where numbers lie in relation to each other. In inequalities like x ≤ 6, a number line illustrates the solution set visually.
For example:

• Locate the number 6 on the number line.
• Use a closed dot at 6, signaling that 6 is part of the solution.
• Shade or point an arrow to the left of 6, covering lesser numbers that satisfy the inequality.

A number line is an excellent way to confirm the boundaries of your solution set, ensuring you visualize what 'less than or equal to' or 'greater than or equal to' encompasses. Understanding how to effectively utilize a number line makes it easier to comprehend solutions intuitively.