Problem 66 - Introductory and Intermediate Algebra for College Students 4th

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 \(x \leq 6\), which includes all real numbers less than or equal to 6.

Step 1: Removing the negative sign on the x variable

To make the inequality easier to solve, the negative sign before the x variable can be removed by adding x to both sides of the inequality: \(3-x+x \geq -3+x\), which simplifies to \(3 \geq -3+x\).

Step 2: Isolate the x variable

To isolate the x variable on one side of the inequality, you can add 3 to both sides to yield \(3+3 \geq -3+3+x\). This further simplifies to \(x \leq 6\).

Step 3: Represent the solution on a number line

The solution \(x \leq 6\) includes all real numbers less than or equal to 6. This can be represented on a number line by marking a filled circle at 6 and drawing a line extending to the left, indicating that all numbers less than or equal to 6 are part of the solution set.

Key Concepts

Addition Property of Inequality

The addition property of inequality is a fundamental principle in solving inequalities. It states that if you add the same number to both sides of an inequality, the direction of the inequality remains unchanged. This property helps us manipulate inequalities without altering their original meaning. For example, in our exercise, we added the variable \(x\) to both sides of the inequality \(3-x \geq -3\). This resulted in \(3 \geq -3 + x\). By doing this, we made the inequality easier to solve by eliminating the negative sign in front of \(x\).

• Remember, whatever you do to one side of the inequality, you must do to the other.
• Addition can help clarify inequalities by simplifying expressions.

Using this property ensures that the inequality’s solution remains valid and accurately describes the relationships between the entities involved.

Multiplication Property of Inequality

Though not explicitly needed for this particular example, the multiplication property of inequality is another essential tool. This property states that when you multiply or divide both sides of an inequality by a positive number, the inequality sign stays the same. However, if you multiply or divide by a negative number, you must reverse the inequality sign.

• This reversal is crucial to maintaining the true nature of the inequality.
• For instance, if \(-x < 5\) and you multiply by \(-1\), it becomes \(x > -5\).

Understanding this property will help you accurately solve and interpret a wide variety of inequality problems. This is particularly useful when you encounter negative coefficients for variables within inequalities.

Graphing Inequalities

Graphing inequalities on a number line provides a visual representation of the solution set. This helps to clearly demonstrate which numbers satisfy the inequality. For the exercise solution of \(x \leq 6\), graphing starts with placing a filled circle on the point 6 on a number line, indicating that 6 is included in the solution.

• The arrow pointing left shows all numbers less than 6 are part of the solution.
• This method helps in understanding the range of values that satisfy the inequality.

Graphing is a powerful tool as it translates mathematical expressions into a visual format that can be quickly interpreted and verified.

Number Line Representation

Using a number line to solve inequalities adds clarity and aids comprehension. When representing an inequality, like \(x \leq 6\), a filled circle on the number line at 6 signifies inclusion of that point in the solution. From this point, a line extends left toward negative infinity to show that every number less than 6 is also a solution.

• A filled circle means the endpoint is included (\(\leq\) or \(\geq\)).
• An open circle means the endpoint is not included (\(<\) or \(>\)).

Number line representation is incredibly useful in verifying and visualizing the solutions of inequalities, making it a valuable method for understanding and solving inequality problems.