Problem 22 - Elementary Algebra Within Reach

Question

Solve and graph the inequality. $$t+1<6$$

Step-by-Step Solution

Answer

The solution to the inequality \(t+1<6\) is \(t<5\). This can be graphed on a number line by marking a point at '5', drawing an open circle (because '5' is not included), and then shading or drawing an arrow to the left to represent all numbers under 5.

Step 1: Simplify the inequality

Simplify the inequality by subtracting 1 from both sides. This gives \(t < 6 - 1\), which simplifies to \(t < 5\)

Step 2: Draw the graph

For graphing, all real numbers that are less than 5 are solutions to the inequality. That means all numbers from 5 onwards to the left on the number line are included. Draw a number line from -10 to 10. Mark '5' on the number line. Draw an open circle at '5', as '5' is not included in the solution of the inequality. Lastly, draw a line to the left to signify all the values less than 5.

Key Concepts

Inequality Solving

Solving linear inequalities is quite similar to solving linear equations. The primary goal is to isolate the variable on one side of the inequality. Let's take our example: \( t+1 < 6 \). A good first step is to simplify the inequality by eliminating any constants from the side of the inequality that includes the variable.

To do this, subtract 1 from both sides to maintain the balance. Doing so, we have \( t+1 - 1 < 6 - 1 \), which simplifies to \( t < 5 \).

Our inequality \( t < 5 \) suggests that any value of \( t \) that is less than 5 is a solution. Remember, when solving inequalities, if you multiply or divide by a negative number, the inequality sign flips direction. However, that's not necessary for this simple inequality.

Graphing Inequalities

Graphing inequalities provides a visual representation of possible solutions on a number line. For instance, for the inequality \( t < 5 \), we need to consider all values of \( t \) that are less than 5.

When we graph it, we start by identifying the critical point on the number line. Here, it's the number 5. Since 5 is not part of the solution (since our inequality shows "less than 5" and not "less than or equal to 5"), we use an **open circle** at 5. An open circle signifies that the number itself is not included in the set of possible solutions.

Next, draw a line extending left from the open circle, indicating that all numbers less than 5 satisfy the inequality. By graphing, we create an intuitive way to view solutions of inequalities.

Number Line Representation

The number line is a powerful tool for representing solutions to inequalities. It helps to visualize not just individual solutions, but an entire set of solutions at a glance.

To effectively represent \( t < 5 \) on a number line:

• Draw a horizontal line which will serve as the number line.
• Mark appropriate intervals covering a range, like from -10 to 10, for good measure.
• Locate and mark the number 5. Place an open circle on this point because 5 itself is not included.
• Draw a line extending to the left from the open circle, showing that all numbers less than 5 are included in the solution set.

By using these steps, anyone can read the graph and understand what values satisfy the inequality.