Problem 16 - Algebra A Combined Function

Question

Solve each inequality. Graph the solution set. Write each answer using solution set notation. $$ -3+m>5 $$

Step-by-Step Solution

Answer

Solution set: \( \{ m \mid m > 8 \} \); graph: open circle at 8, shade right.

Step 1: Isolate the variable

Start solving the inequality by isolating the variable on one side. In the inequality \(-3 + m > 5\), we aim to isolate \(m\). To do this, add \(3\) to both sides of the inequality:\[-3 + m + 3 > 5 + 3\] Simplifying this gives:\[m > 8\] So, the inequality simplifies to \(m > 8\).

Step 2: Graph the solution

To graph the solution \(m > 8\) on a number line, first draw a number line. Place an open circle at 8 to indicate that 8 is not included in the solution. Then, shade to the right of the 8 to show all numbers greater than 8.

Step 3: Write the solution set

The solution set represents all values that satisfy \(m > 8\). In solution set notation, express this as: \( \{ m \mid m > 8 \} \), meaning 'the set of all \(m\) such that \(m\) is greater than \(8\)'.

Key Concepts

Solution Set Notation

Solution set notation is a way to describe the set of all possible solutions to an inequality in a concise manner. It is particularly useful in expressing inequalities that have multiple solutions. Take the inequality from our example, where we found that \(m > 8\). To express this in solution set notation, we write it as:

• \( \{ m \mid m > 8 \} \)

This translates to "the set of all \(m\) such that \(m\) is greater than 8." It's a neat way to convey the range of values satisfying the inequality, and it uses a vertical bar \(|\) to mean "such that."
Mastering solution set notation helps in clearly presenting the solutions to inequalities, which is fundamental for understanding how inequalities encompass a range rather than a single value.

Graphing Inequalities

Graphing inequalities is a visual method to depict the solution to an inequality on a number line. Let's see how to graph the inequality \(m > 8\). Begin by drawing a simple horizontal line representing the number line.

• Identify the number 8 on this line. Since the inequality is \(m > 8\), place an open circle on 8. The open circle indicates that 8 itself is not a part of the solution.
• The next step is to shade or draw an arrow to the right of the circle. This indicates all numbers greater than 8 are part of the solution.

Graphing provides a clear visual of the inequality's range and immediately shows which numbers satisfy the condition. Understanding how to effectively graph inequalities significantly aids in comprehending the scope of the solution and enables quick checks for accuracy.

Solving Inequalities Step-by-Step

Solving inequalities step-by-step involves several key actions to isolate the variable, similar to solving equations. Let's walk through the process with the example \(-3 + m > 5\):

• Step 1: Isolate the variable. Our goal is to rearrange the inequality so that \(m\) is on one side by itself. Begin by adding 3 to both sides, ensuring the inequality remains balanced: \(-3 + m + 3 > 5 + 3\).
• Simplify this to achieve: \(m > 8\).

Remember that the rules for manipulating inequalities closely mirror those for equations, except when multiplying or dividing by a negative number, which reverses the inequality symbol. By following these steps methodically, you can solve any inequality with confidence.