Problem 19 - Algebra 2

Question

Solve each inequality. Then graph the solution set on a number line. \(6 b+11 \geq 15\)

Step-by-Step Solution

Answer

The solution is \(b \geq \frac{2}{3}\); graph with a closed circle at \(\frac{2}{3}\) and shade to the right.

Step 1: Isolate the variable term

Start by isolating the term with the variable on one side of the inequality. Begin by subtracting 11 from both sides to eliminate the constant term on the left-hand side: \[6b + 11 - 11 \geq 15 - 11\]This simplifies to:\[6b \geq 4\]

Step 2: Solve for the variable

To solve for \(b\), divide both sides of the inequality by 6 to get \(b\) by itself:\[\frac{6b}{6} \geq \frac{4}{6}\]This simplifies to:\[b \geq \frac{2}{3}\]

Step 3: Graph the solution

To graph the inequality \(b \geq \frac{2}{3}\) on a number line, draw a number line with a closed circle at \(\frac{2}{3}\), indicating that \(\frac{2}{3}\) is included in the solution. Shade the line to the right of \(\frac{2}{3}\) to show that \(b\) can be any number greater than or equal to \(\frac{2}{3}\).

Key Concepts

Graphing Inequalities

Graphing inequalities involves showing the range of possible values that satisfy the inequality on a visual graph. This can help provide a clearer understanding of how the inequality behaves. In our exercise, we started with the inequality \(6b + 11 \geq 15\) and simplified it to \(b \geq \frac{2}{3}\). The goal is to visualize this condition on a number line.

When graphing an inequality, the symbol of the inequality dictates the type of circle we use when plotting on the number line. Here are the general rules:

• \(>\) or \(<\): Use an open circle to show that the number itself is not included.
• \(\geq\) or \(\leq\): Use a closed circle to indicate that the number itself is included in the solution set.

In this case, since we have \(b \geq \frac{2}{3}\), a closed circle is drawn at \(\frac{2}{3}\), and we shade the line to the right, indicating all values that satisfy the inequality.

Inequality Solutions

Solving an inequality involves finding the set of values that make the inequality true. This step is similar to solving equations, but keep in mind the rules specific to inequalities. Let’s break it down using our example.

The inequality \(6b + 11 \geq 15\) requires us to isolate \(b\). We do this by strategically performing operations on both sides of the inequality. It starts with subtracting 11, simplifying it, and then dividing by 6, to reveal that \(b\) must be greater than or equal to \(\frac{2}{3}\).

Here's a quick guide on handling inequalities:

• Avoid changing the inequality's direction unless multiplying or dividing by a negative number.
• Work step-by-step, just like in equation solving.
• The final solution, \(b \geq \frac{2}{3}\), gives us a range of values that satisfy the condition.

Once the solution is found, it can be interpreted graphically on a number line for better understanding.

Number Line Representation

A number line is a visual tool often used to display the solution of inequalities. It helps in understanding the range of possible values that forms the solution set. In our example of \(b \geq \frac{2}{3}\), the number line provides an intuitive way to visualize this.

Here's how you can represent inequalities like \(b \geq \frac{2}{3}\) on a number line:

• Determine the critical value: Begin with the crucial number, like \(\frac{2}{3}\), in this case.
• Draw the number line: Add a point for the critical value. Use a closed circle because the inequality includes this value (since it’s \(\geq\)).
• Indicate the solution set: Shade or draw an arrow to the right of \(\frac{2}{3}\) to show all possible values greater than or equal to \(\frac{2}{3}\).

This representation simplifies understanding which values \(b\) can take, making inequalities easier to interpret and solve for anyone learning the concept.