Problem 57 - 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. \(2 x-3>7\)

Step-by-Step Solution

Answer

The solution to the inequality \(2x - 3 > 7\) is \(x > 5\). On the number line, this is represented by a circle above the 5, with shading to the right of the circle.

Step 1: Add 3 to both sides

Start by getting rid of the -3 on the left-hand side. We do this by adding 3 to both sides of the inequality. This operation gives us \(2x > 10\) since \(7 + 3 = 10\).

Step 2: Divide by 2

To isolate the variable x, we need to remove the 2 that it's currently being multiplied by. We do this by dividing each side of our inequality by 2. This gives us \(x>5\), as \(10 ÷ 2 = 5\). This is the solution to the inequality.

Step 3: Plotting the solution on a number line

First draw a straight line. Then, put a small circle above the 5 and shade to the right of this circle, because x is greater than 5. All points to the right of 5 on the number line are solutions to the inequality.

Key Concepts

Addition Property of Inequality

Understanding the addition property of inequality is fundamental when solving inequalities. This property states that you can add the same number to both sides of an inequality without changing its direction. Think about it as keeping the balance of a scale; if you add weight to both sides equally, the heavier side remains heavier.

For example, considering the inequality in our exercise, we started with the expression \(2x - 3 > 7\). To move towards isolating the variable \(x\), we added 3 to both sides of the inequality, resulting in \(2x > 10\). It is essential to remember that this addition property is valid for both positive and negative numbers, and what we do to one side, we must do to the other to maintain the balance of the inequality.

Multiplication Property of Inequality

Just as addition is a key step, the multiplication property of inequality is another tool in our algebraic toolbox. This property indicates that multiplying or dividing both sides of an inequality by a positive number will not change the direction of the inequality. Again, it's all about balance—like adjusting the weights on a balanced scale without tipping it over.

From our example, after using the addition property, we had \(2x > 10\). The next step was to isolate \(x\) by getting rid of the 2 that was multiplying it. We achieved this by dividing both sides by 2, which gave us \(x > 5\). It's crucial to note that if we were to multiply or divide by a negative number, the inequality symbol would need to be reversed to maintain the correct balance.

Graphing Solutions on a Number Line

Once we have solved the inequality, graphing the solution on a number line helps us visualize the range of possible values. A number line is a straight, horizontal line with numbers placed at intervals, which can represent the set of all real numbers.

The way we graph an inequality depends on the relationship it describes. In our exercise, we found that \(x > 5\). To represent this on a number line, we draw an open circle at 5 to indicate that 5 is not included in the solution set. Then, we shade or draw an arrow to the right of 5, extending towards the larger numbers, representing all the values that are greater than 5. This visual representation demonstrates the set of all possible solutions in a clear and effective manner.

Isolation of Variables

The goal when dealing with an inequality is to isolate the variable in question, which means getting the variable by itself on one side of the inequality. The process typically involves using the addition and multiplication properties of inequality, as we've discussed.

In the given exercise, our variable \(x\) was initially accompanied by -3 and multiplied by 2, which means we needed to perform operations that would undo these combinations. First, we added 3 to cancel out the -3, and then we divided by 2 to counteract the multiplication by 2. Isolating the variable renders a clear statement regarding its potential values and is the final step to solving an inequality before representing it on a number line.