Problem 2 - Prealgebra

Question

Solve each inequality and check your solution. Then graph the solution on a number line. $$12 a-4>20$$

Step-by-Step Solution

Answer

The solution is \(a > 2\). Graphically, it is an open circle at 2 with shading to the right.

Step 1 - Isolate the variable term

Start by isolating the term with the variable. Add 4 to both sides of the inequality:\[12a - 4 + 4 > 20 + 4\]Which simplifies to:\[12a > 24\]

Step 2 - Solve for the variable

Divide both sides of the inequality by 12 to solve for \(a\):\[\frac{12a}{12} > \frac{24}{12}\]This simplifies to:\[a > 2\]

Step 3 - Check the solution

Choose a value for \(a\) that is greater than 2, such as 3, and substitute it back into the original inequality to verify:\[12(3) - 4 > 20\]Simplify the left side:\[36 - 4 > 20\]\[32 > 20\]Since this is true, the solution is verified.

Step 4 - Graph the solution

To graph \(a > 2\) on a number line, draw a number line with an open circle at 2 (since 2 is not included) and shade everything to the right of 2, indicating all numbers greater than 2 as part of the solution.

Key Concepts

Graphing Inequalities

Graphing inequalities is a way to visually represent solutions to inequality expressions. When you have an inequality like \(a > 2\), this means that the solution includes every number greater than 2 but not 2 itself. On a graph, this can be shown using a number line.

When graphing inequalities on a number line, follow these steps:

• Identify the boundary number, which is 2 in our example.
• Decide if the boundary will be an open or closed circle. Since it’s \(a > 2\), we use an open circle at 2 to show that 2 is not part of the solution.
• Shade all the numbers to the right of the open circle to represent all numbers greater than 2.

Understanding how to graph inequalities helps you quickly interpret the solutions. You can see at a glance which numbers make the inequality true and which do not.

Algebraic Expressions

An algebraic expression comprises variables, constants, and arithmetic operations. In solving inequalities, you manipulate these expressions to isolate the variable.

Let's look at the algebraic expression in the example: \(12a - 4 > 20\). Your initial goal is to isolate the variable term (in this case, \(a\)) to find its value.

Here’s how you do it:

• Add or subtract terms to move constants from one side of the inequality to the other, like moving \(-4\) to the right by adding 4 to both sides.
• Perform multiplication or division to solve for the variable. Divide each term by 12 to simplify \(12a > 24\) to \(a > 2\).

Understanding algebraic expressions and how to manipulate them is essential for solving inequalities, allowing you to systematically approach finding solutions.

Number Line Representation

Using a number line to represent solutions is an effective way to visualize inequalities. The number line acts as a spectrum of possible values that make the inequality true.

To accurately represent \(a > 2\) on a number line:

• Draw a horizontal line and mark relevant points, such as the boundary number 2.
• Place an open circle above 2 to show it's not included. The open circle visually denotes exclusion.
• Shade to the right of the open circle to indicate that every number greater than 2 is part of the solution set. This shading visually communicates the range of valid solutions.

Number line representation is crucial because it provides a clear, visual understanding of inequalities, making them easier to interpret and ensuring both accuracy and clarity in communication.