Problem 22 - 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. \(x+1<6\)

Step-by-Step Solution

Answer

'x' is less than 5 and this solution is represented on the number line by an open circle at '5' with a line or arrow extending to the left.

Step 1: Apply Addition Property of Inequality

The given inequality is \(x + 1 < 6\). We need to isolate 'x'. To do this, we can subtract 1 from both sides of the inequality using the addition property of inequality to maintain the balance; doing so, we get \(x + 1 - 1 < 6 - 1\). This simplifies to \(x < 5\). That is, 'x' is less than 5.

Step 2: Graphing on a Number Line

Next, we need to graph 'x < 5' on a number line. Draw a number line and mark '5' on it. Since 'x' is strictly less than '5', we put an open circle at '5'. Then, we shade or point an arrow to the left of '5' indicating that 'x' can be any number less than '5'.

Key Concepts

Addition Property of Inequality

When solving inequalities, the Addition Property of Inequality is a crucial tool. It allows you to add (or subtract) the same number from both sides of an inequality without changing the inequality's direction. This helps in isolating the variable we are interested in.
For instance, in the inequality \( x + 1 < 6 \), we subtracted 1 from both sides. This gives us \( x < 5 \). By applying this property, we've successfully isolated \( x \), and the inequality remains intact.
In essence, remember:

• If you add or subtract the same number from both sides of the inequality, the direction of the inequality symbol does not change.
• This is similar to solving equations except instead of an equality sign, you maintain the inequality sign.

Mastering how to manipulate inequalities using this property underpins effective inequality solving.

Graphing Inequalities

Graphing inequalities helps visualize the range of values that solve the inequality. It shows the solution set right on the number line.
To graph \( x < 5 \), you start by drawing a number line and marking the number 5. With graphing inequalities, the type of circle used (open or closed) signifies whether a number is included in the solution set.

• An open circle is used for inequalities like \( < \) or \( > \), implying the number itself isn't part of the solution.
• A closed circle is used for \( \leq \) or \( \geq \), to indicate that the number is included in the solution.

In this case, because our inequality is \( x < 5 \), we use an open circle at 5. This tells us 5 is not a solution, but all numbers less than 5 are.
Understanding this graphical representation makes it easier to comprehend which values satisfy the inequality.

Number Line Representation

A number line is an excellent visual tool for representing solutions of inequalities. It offers an immediate and clear depiction of the set of possible solutions.
To effectively use a number line:

• Ensure the appropriate numbers are marked, focusing on the critical point, in this case, 5.
• Decide on the type of circle (open or closed) based on whether the boundary number is included.

After marking 5 with an open circle, shade the region to the left of 5. This shaded area represents all possible solutions \( (x < 5) \).
When learning or teaching inequalities, visually representing them on a number line can enhance understanding greatly. It delineates the boundaries clearly and visually distinguishes which numbers make the inequality true.