Problem 11 - Algebra 2

Question

Solve each inequality. Then graph the solution set on a number line. \(b-3 \leq 15\)

Step-by-Step Solution

Answer

\(b \leq 18\), graph as a filled dot at 18 with a line extending left.

Step 1: Isolate the Variable

To solve the inequality \(b - 3 \leq 15\), we need to isolate the variable \(b\). We do this by adding 3 to both sides of the inequality: \(b - 3 + 3 \leq 15 + 3\). Simplifying this gives us \(b \leq 18\).

Step 2: Represent on a Number Line

To graph the solution \(b \leq 18\) on a number line, we draw a number line and locate the point 18 on it. Since 18 is included in the solution set (indicated by the \(\leq\) sign), we draw a filled dot (or circle) at 18. Then, shade the line extending to the left of 18 to indicate all numbers less than or equal to 18 are included in the solution set.

Key Concepts

Number Line Representation

A number line is a simple tool used to visualize the set of solutions for an inequality. It helps us understand what numbers satisfy the condition given by the inequality. In our problem, once we have determined that the solution is all numbers less than or equal to 18, we can use a number line to depict this:

• Draw a horizontal line and mark equal intervals on it.
• Locate the number 18 on this line.
• Because the inequality involves "less than or equal to 18" (\(\leq\)), we use a filled dot at 18. This filled dot indicates that 18 itself is a part of the solution.
• Shade or draw an arrow from the dot at 18 extending to the left to signify all numbers smaller than 18 are included.

By representing inequalities on a number line, you make it easier to visually confirm the correct set of solutions.

Isolating the Variable

Isolating the variable is a crucial step in solving inequalities. It means rearranging the inequality to have the variable on one side by itself. Here's how it works in the example:

• You start with: \(b - 3 \leq 15\).
• Your goal is to have \(b\) by itself on one side of the inequality.
• To do this, perform the opposite operation to isolate \(b\). Here, subtracting 3 is the current operation, so you will add 3 to both sides.
• When you add 3: \(b - 3 + 3 \leq 15 + 3\), simplifying gives \(b \leq 18\).

When isolating variables, always ensure you do the same operation to both sides of the inequality. This maintains balance, ensuring the inequality stays true after manipulation.

Graphical Solution of Inequalities

Using graphs to solve inequalities provides a visual component to understanding their solutions. When you solve an inequality, you are often interested in knowing how all the solutions are related and how they can be visually represented. In our example:

• We determined the solution to be all numbers \(b\) such that \(b \leq 18\).
• Graphically, these solutions are shown on the number line, which, as discussed, simplifies comprehension of which numbers satisfy the inequality.

Graphically solving inequalities is not only helpful in verifying your solution but also in providing insight into the behavior of solutions. The visual representation shows the range of values satisfying the inequality, making complex solutions more intuitive.