Graph The Solution Of The Inequality On The Number Line.$t - 3 \leq 2$

Introduction

In mathematics, inequalities are used to compare the values of two or more expressions. Graphing the solution of an inequality on a number line is a useful tool for visualizing the solution set and understanding the relationship between the variables. In this article, we will focus on graphing the solution of the inequality $t - 3 \leq 2$ on the number line.

Understanding the Inequality

The given inequality is $t - 3 \leq 2$. To graph the solution on the number line, we need to understand the inequality and its components. The inequality is in the form of $x \leq y$, where $x$ is the variable $t$ and $y$ is the constant $2$. The inequality states that the value of $t$ is less than or equal to $2$.

Graphing the Solution

To graph the solution of the inequality on the number line, we need to follow these steps:

1. Identify the critical point: The critical point is the value of $t$ that makes the inequality true. In this case, the critical point is $t = 2$.
2. Determine the direction of the inequality: Since the inequality is in the form of $x \leq y$, the solution set will include all values of $t$ that are less than or equal to $2$.
3. Graph the solution set: To graph the solution set, we will use a closed circle to represent the critical point $t = 2$. We will then draw a line to the left of the critical point to represent all values of $t$ that are less than or equal to $2$.

Graphing the Solution Set

The graph of the solution set is shown below:

• Critical point: $t = 2$
• Direction of the inequality: $\leq$
• Graph of the solution set: A closed circle at $t = 2$ and a line to the left of the critical point.

Interpretation of the Graph

The graph of the solution set represents all values of $t$ that satisfy the inequality $t - 3 \leq 2$. The closed circle at $t = 2$ represents the critical point, and the line to the left of the critical point represents all values of $t$ that are less than or equal to $2$.

Conclusion

Graphing the solution of an inequality on the number line is a useful tool for visualizing the solution set and understanding the relationship between the variables. By following the steps outlined in this article, we can graph the solution of the inequality $t - 3 \leq 2$ on the number line. The graph of the solution set represents all values of $t$ that satisfy the inequality and provides a clear understanding of the relationship between the variables.

Example Problems

1. Graph the solution of the inequality $x + 2 \geq 5$ on the number line.
2. Graph the solution of the inequality $y - 1 \leq 3$ on the number line.
3. Graph the solution of the inequality $z + 4 \leq 2$ on the number line.

Solutions

1. Graph the solution of the inequality $x + 2 \geq 5$ on the number line.

   • Critical point: $x = 3$
   • Direction of the inequality: $\geq$
   • Graph of the solution set: A closed circle at $x = 3$ and a line to the right of the critical point.

2. Graph the solution of the inequality $y - 1 \leq 3$ on the number line.

   • Critical point: $y = 4$
   • Direction of the inequality: $\leq$
   • Graph of the solution set: A closed circle at $y = 4$ and a line to the left of the critical point.

3. Graph the solution of the inequality $z + 4 \leq 2$ on the number line.

   • Critical point: $z = -2$
   • Direction of the inequality: $\leq$
   • Graph of the solution set: A closed circle at $z = -2$ and a line to the left of the critical point.

Tips and Tricks

• Use a closed circle to represent the critical point: A closed circle represents a value that is included in the solution set.
• Use a line to represent the direction of the inequality: A line represents the direction of the inequality and indicates whether the solution set includes values greater than, less than, or equal to the critical point.
• Graph the solution set carefully: Make sure to graph the solution set accurately and clearly, and include all values that satisfy the inequality.

Conclusion

Introduction

In the previous article, we discussed how to graph the solution of an inequality on the number line. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on graphing the solution of an inequality on the number line.

Q&A

Q1: What is the critical point in an inequality?

A1: The critical point is the value of the variable that makes the inequality true. It is the point at which the inequality changes direction.

Q2: How do I determine the direction of the inequality?

A2: To determine the direction of the inequality, look at the inequality sign. If the inequality sign is $\leq$, the solution set includes all values less than or equal to the critical point. If the inequality sign is $\geq$, the solution set includes all values greater than or equal to the critical point.

Q3: What is the difference between a closed circle and an open circle on the number line?

A3: A closed circle represents a value that is included in the solution set, while an open circle represents a value that is not included in the solution set.

Q4: How do I graph the solution set on the number line?

A4: To graph the solution set, use a closed circle to represent the critical point and a line to represent the direction of the inequality. If the inequality sign is $\leq$, draw a line to the left of the critical point. If the inequality sign is $\geq$, draw a line to the right of the critical point.

Q5: What if the inequality has multiple critical points?

A5: If the inequality has multiple critical points, graph each critical point separately and draw a line to represent the direction of the inequality. Make sure to include all values that satisfy the inequality.

Q6: How do I graph the solution of an inequality with a variable on both sides?

A6: To graph the solution of an inequality with a variable on both sides, first simplify the inequality by combining like terms. Then, graph the solution set using the steps outlined in the previous article.

Q7: What if the inequality has a fraction or a decimal?

A7: If the inequality has a fraction or a decimal, simplify the fraction or decimal to a whole number or a decimal with a finite number of digits. Then, graph the solution set using the steps outlined in the previous article.

Q8: How do I graph the solution of an inequality with absolute value?

A8: To graph the solution of an inequality with absolute value, first simplify the absolute value expression. Then, graph the solution set using the steps outlined in the previous article.

Q9: What if the inequality has a negative sign?

A9: If the inequality has a negative sign, change the direction of the inequality sign. For example, if the inequality is $x - 3 \geq 2$, change the direction of the inequality sign to $x - 3 \leq 2$.

Q10: How do I graph the solution of an inequality with multiple variables?

A10: To graph the solution of an inequality with multiple variables, graph each variable separately and then combine the graphs to represent the solution set.

Conclusion

Graphing the solution of an inequality on the number line is a useful tool for visualizing the solution set and understanding the relationship between the variables. By following the steps outlined in this article and using the Q&A section, we can graph the solution of any inequality on the number line.

Example Problems

1. Graph the solution of the inequality $x + 2 \geq 5$ on the number line.
2. Graph the solution of the inequality $y - 1 \leq 3$ on the number line.
3. Graph the solution of the inequality $z + 4 \leq 2$ on the number line.

Solutions

1. Graph the solution of the inequality $x + 2 \geq 5$ on the number line.

   • Critical point: $x = 3$
   • Direction of the inequality: $\geq$
   • Graph of the solution set: A closed circle at $x = 3$ and a line to the right of the critical point.

2. Graph the solution of the inequality $y - 1 \leq 3$ on the number line.

   • Critical point: $y = 4$
   • Direction of the inequality: $\leq$
   • Graph of the solution set: A closed circle at $y = 4$ and a line to the left of the critical point.

3. Graph the solution of the inequality $z + 4 \leq 2$ on the number line.

   • Critical point: $z = -2$
   • Direction of the inequality: $\leq$
   • Graph of the solution set: A closed circle at $z = -2$ and a line to the left of the critical point.

Tips and Tricks

• Use a closed circle to represent the critical point: A closed circle represents a value that is included in the solution set.
• Use a line to represent the direction of the inequality: A line represents the direction of the inequality and indicates whether the solution set includes values greater than, less than, or equal to the critical point.
• Graph the solution set carefully: Make sure to graph the solution set accurately and clearly, and include all values that satisfy the inequality.