Problem 59 - 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. \(3 x+3<18\)

Step-by-Step Solution

Answer

The solution to the inequality \(3x + 3 < 18\) is \(x < 5\).

Step 1: Use the Addition Property of Inequality

Subtract 3 from both sides of the inequality to isolate the term with x. This leads to \(3x < 18 - 3\)

Step 2: Simplify the Right-Hand Side

Simplify the expression on the right-hand side to get \(3x < 15\).

Step 3: Use the Multiplication Property of Inequality

Divide both sides of the inequality by 3 to isolate x. This gives \(x < 15/3\).

Step 4: Simplify the Right-Hand Side

Simplify the expression on the right-hand side to get \(x < 5\).

Step 5: Graph the Solution Set on a Number Line

Draw a number line, mark the point x = 5, and shade everything to its left because x < 5. The point x = 5 is not included in the solution set.

Key Concepts

Addition Property of Inequality

When you're solving inequalities, you need to know about the addition property of inequality. This is very similar to solving regular equations. You can add or subtract the same number from both sides of an inequality without changing its direction. This property is helpful because it allows us to isolate terms and simplify the inequality while maintaining its truth.

For instance, in the inequality \(3x + 3 < 18\), we want to isolate \(3x\). By using the addition property, we subtract 3 from both sides. This simplifies the inequality to \(3x < 15\). By doing this, you ensure everything stays balanced, just like a seesaw that remains even when you remove equal weights from both sides.

• Keep the inequality balanced by performing the same operation on both sides.
• The direction of the inequality remains the same when using addition or subtraction.
• This step is crucial for simplifying inequalities for further solving.

Multiplication Property of Inequality

Another important tool is the multiplication property of inequality. It allows you to multiply or divide both sides of an inequality by a positive number without reversing the inequality sign. However, remember, if you multiply or divide by a negative number, you must flip the inequality sign.

In our example \(3x < 15\), we want to find out what \(x\) is on its own. So, we divide both sides by 3. Since 3 is positive, the inequality sign stays the same. This results in \(x < 5\).

• When multiplying or dividing by a positive number, the inequality sign stays the same.
• Always reverse the inequality sign when multiplying or dividing by a negative number.
• This step is about simplifying the variable's coefficient to one for clear comparison.

Graphing Solution Sets

Once you've solved your inequality, it's effective to visualize the solution on a number line. This shows all the possible values that satisfy the inequality. In the inequality \(x < 5\), we draw a number line and focus on the number 5.

To graph \(x < 5\):
- Put an open circle on 5. This indicates that \(x = 5\) is not part of the solution, as the inequality is strictly less than. - Shade all the numbers to the left of 5. This shading represents all numbers less than 5 that satisfy the inequality.

• An open circle is used for inequalities that do not include the number (\(\lt\) or \(\gt\)).
• A closed circle is used when the number is included (\(\leq\) or \(\geq\)).
• Shading the line on the appropriate side shows the range of values that make the inequality true.