Problem 10 - Prealgebra

Question

Solve each inequality and check your solution. Then graph the solution on a number line. $$2 x+8>24$$

Step-by-Step Solution

Answer

The solution is \(x > 8\). Graph shows values greater than 8.

Step 1: Isolate the Variable

To solve the inequality \(2x + 8 > 24\), we first need to isolate the variable \(x\). Start by subtracting 8 from both sides of the inequality to get \(2x > 16\).

Step 2: Solve for x

Next, divide both sides of the inequality by 2 to solve for \(x\). This gives us \(x > 8\).

Step 3: Check the Solution

Choose a test value greater than 8, such as 9, and substitute it back into the original inequality to ensure it holds true: \(2(9) + 8 = 26 > 24\). Therefore, our solution \(x > 8\) is correct.

Step 4: Graph the Solution

To graph the solution \(x > 8\) on a number line, draw a number line with a circle at 8 and shade the region to the right of 8, indicating all numbers greater than 8 are solutions.

Key Concepts

Isolating Variables

When solving inequalities, the first step is to isolate the variable. This means you want the variable by itself on one side of the inequality sign, such as in "greater than," "less than," etc. Let's apply this to our example inequality: 2x + 8 > 24.

• First, subtract 8 from both sides: \(2x + 8 - 8 > 24 - 8\)
• This simplifies to: \(2x > 16\)
• Next, divide both sides by 2 to get the variable x by itself: \(\frac{2x}{2} > \frac{16}{2}\)
• So, the inequality simplifies to: \(x > 8\).

Isolating the variable helps you determine the range of values that satisfy the inequality.

Checking Solutions

Checking your solution is crucial to ensure your inequality was solved correctly. It involves substituting a value from the solution set back into the original inequality.

• First, select a number greater than 8. For instance, let’s pick 9.
• Substitute it into the original inequality: \(2(9) + 8 = 26\).
• Check the result to see if it satisfies the inequality: \(26 > 24\).

The inequality holds true, confirming that our solution \(x > 8\) is correct. Checking solutions helps catch any mistakes and verifies the accuracy of your answers.

Graphing Inequalities

Once you've found the solution to an inequality, it's helpful to represent it visually by graphing it. This provides a clear, intuitive understanding of the set of solutions.

• Start by drawing a number line, a horizontal line with numbers marked at regular intervals.
• Then, locate the significant point, in this case, 8, and draw an open circle on it.
• An open circle is used here because the inequality \(x > 8\) doesn't include 8 itself.
• Shade the area to the right of this circle. The shading represents all numbers that are solutions to the inequality, which are greater than 8.

Graphing inequalities not only aids in confirming solutions but also in visually interpreting the possible values that satisfy the inequality.

Number Line

A number line is a straight, horizontal line that coordinates numbers at regular intervals. It's a fantastic tool for graphing inequalities as it visually represents solution sets neatly.

• Let’s consider our solution \(x > 8\). Draw a straight horizontal line and label it with numbers, marking them at equal distances.
• Identify the point where x equals 8 and mark it with an open circle to signify that 8 is not included in the solution.
• Shade or draw an arrow to the right. This represents all numbers greater than 8, which are included in our solution set.

Using a number line makes it easy to see which numbers satisfy the inequality, reinforcing the understanding of inequalities' solutions.