Problem 28 - Introductory Algebra for College Students

Question

Use the addition property of inequality to solve each inequality and graph the solution set on a number line. \(2 x+9 \leq x+2\)

Step-by-Step Solution

Answer

The solution to the inequality is \(x \leq -7\), which can be represented on a number line by shading all numbers less than or equal to -7.

Step 1: Isolate the variable x

First, we subtract x from both sides to get the inequality \(x + 9 \leq 2\). This separates x on the left side of the inequality.

Step 2: Solve for x

Then subtract 9 from both sides of the inequality to isolate x completely. This results in \(x \leq -7\).

Step 3: Graph the solution

The inequality \(x \leq -7\) means that x is any number that is less than or equal to -7. This can be represented on a number line by shading all numbers to the left of and including -7. A closed circle is placed on -7 to indicate that -7 is part of the solution set.

Key Concepts

Addition Property of Inequality

The addition property of inequality is a foundational concept that helps simplify and solve inequalities. It states that you can add the same number to both sides of an inequality without changing its truth. This property is useful for isolating variables.

For instance, in the inequality \(2x + 9 \leq x + 2\), we use this property by subtracting \(x\) from both sides. By treating subtraction as adding a negative, we maintain the inequality's balance.

• Start with the inequality: \(2x + 9 \leq x + 2\).
• Subtract \(x\) from both sides: \(x + 9 \leq 2\).

Always pay attention to the direction of the inequality. It will not change unless multiplying or dividing by a negative number, but here, since we subtracted the elements, the inequality remains in the same direction.

Solution Set

The solution set of an inequality includes all the values of the variable that make the inequality true. It's essential to represent these values correctly to get the complete picture of possible solutions.

After simplifying the inequality \(x + 9 \leq 2\) to \(x \leq -7\), all numbers less than or equal to -7 comprise the solution set. Always express the result clearly for better understanding, such as:

• If the inequality is \(x \leq -7\), then the solution set is "all real numbers that are at most -7."
• This means every number from negative infinity up to and including -7.

This interpretation helps not only in solving more complex inequalities but also supports graph visualization on a number line.

Number Line

Visualizing inequalities on a number line provides a clear representation of the solution set, where values are neatly displayed in one-dimensional space.

To graph \(x \leq -7\) on a number line:

• Draw a horizontal line and mark key values, like -7, on it.
• Place a closed circle on -7 to signify it is included in the solution set. A closed circle means the endpoint is part of the set.
• Shade all numbers to the left of -7, indicating that every number less than -7 is also part of the solution.

Number lines make inequalities more tangible, especially visual learners. They also aid in checking the logic of your solution by providing an immediate view of what is included.