Which Point Is NOT Part Of The Solution Of The Inequality $y \ \textless \ |3x| + 1$?A. (-4, 12)B. (2, -9)C. (3, 15)D. (0, 0)

Introduction

Inequalities are mathematical expressions that compare two values, often with a greater-than or less-than symbol. Solving inequalities involves finding the values of the variable that satisfy the given inequality. In this article, we will focus on solving the inequality $y \ \textless \ |3x| + 1$ and identify the point that is NOT part of the solution.

Understanding Absolute Value

The absolute value of a number is its distance from zero on the number line. For example, the absolute value of 3 is 3, and the absolute value of -3 is also 3. The absolute value function is denoted by two vertical lines, such as $|x|$. When we have an absolute value in an inequality, we need to consider both the positive and negative cases.

Solving the Inequality

To solve the inequality $y \ \textless \ |3x| + 1$, we need to consider two cases:

• Case 1: $3x \geq 0$
• Case 2: $3x < 0$

Case 1: $3x \geq 0$

When $3x \geq 0$, we can remove the absolute value sign and rewrite the inequality as $y \ \textless \ 3x + 1$. To solve this inequality, we can use the following steps:

1. Subtract 1 from both sides: $y - 1 \ \textless \ 3x$
2. Divide both sides by 3: $\frac{y - 1}{3} \ \textless \ x$

Case 2: $3x < 0$

When $3x < 0$, we need to consider the negative case of the absolute value. We can rewrite the inequality as $y \ \textless \ -3x + 1$. To solve this inequality, we can use the following steps:

1. Subtract 1 from both sides: $y - 1 \ \textless \ -3x$
2. Divide both sides by -3: $\frac{y - 1}{-3} \ \textgreater \ x$

Combining the Cases

To find the solution to the original inequality, we need to combine the two cases:

• For $3x \geq 0$, we have $\frac{y - 1}{3} \ \textless \ x$
• For $3x < 0$, we have $\frac{y - 1}{-3} \ \textgreater \ x$

Graphing the Solution

To visualize the solution, we can graph the two inequalities on a coordinate plane. The solution to the original inequality is the region below the graph of $y = |3x| + 1$.

Identifying the Point

Now that we have the solution to the inequality, we can identify the point that is NOT part of the solution. Let's examine the answer choices:

• A. (-4, 12)
• B. (2, -9)
• C. (3, 15)
• D. (0, 0)

Analyzing the Answer Choices

To determine which point is NOT part of the solution, we need to check if each point satisfies the inequality. We can substitute the x and y values of each point into the inequality and check if it is true.

• A. (-4, 12): $12 \ \textless \ |3(-4)| + 1$ is true, so this point is part of the solution.
• B. (2, -9): $-9 \ \textless \ |3(2)| + 1$ is false, so this point is NOT part of the solution.
• C. (3, 15): $15 \ \textless \ |3(3)| + 1$ is true, so this point is part of the solution.
• D. (0, 0): $0 \ \textless \ |3(0)| + 1$ is true, so this point is part of the solution.

Conclusion

In conclusion, the point that is NOT part of the solution to the inequality $y \ \textless \ |3x| + 1$ is B. (2, -9). This point does not satisfy the inequality, and therefore, it is not part of the solution.

Final Thoughts

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Q: What is an inequality?

A: An inequality is a mathematical expression that compares two values, often with a greater-than or less-than symbol.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable on one side of the inequality sign. You can do this by adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.

Q: What is absolute value?

A: Absolute value is the distance of a number from zero on the number line. For example, the absolute value of 3 is 3, and the absolute value of -3 is also 3.

Q: How do I handle absolute value in an inequality?

A: When you have an absolute value in an inequality, you need to consider both the positive and negative cases. You can remove the absolute value sign and rewrite the inequality as two separate inequalities.

Q: How do I graph the solution to an inequality?

A: To graph the solution to an inequality, you can use a coordinate plane. The solution to the inequality is the region below the graph of the inequality.

Q: What is the difference between a linear inequality and a quadratic inequality?

A: A linear inequality is an inequality that can be written in the form $ax + b \ \textless \ c$, where $a$, $b$, and $c$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c \ \textless \ 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you can use the following steps:

1. Factor the quadratic expression, if possible.
2. Set each factor equal to zero and solve for the variable.
3. Use a number line or a graph to determine the intervals where the inequality is true.

Q: What is the importance of understanding inequalities?

A: Understanding inequalities is important in many areas of mathematics, science, and engineering. Inequalities are used to model real-world problems, such as optimization problems, and to make predictions about the behavior of systems.

Q: How can I practice solving inequalities?

A: You can practice solving inequalities by working through examples and exercises in a textbook or online resource. You can also try solving inequalities on your own, using a calculator or computer program to check your answers.

Q: What are some common mistakes to avoid when solving inequalities?

A: Some common mistakes to avoid when solving inequalities include:

• Not isolating the variable on one side of the inequality sign.
• Not considering both the positive and negative cases when handling absolute value.
• Not using a number line or graph to determine the intervals where the inequality is true.
• Not checking your answers using a calculator or computer program.

Q: How can I apply inequalities to real-world problems?

A: Inequalities can be used to model real-world problems, such as optimization problems, and to make predictions about the behavior of systems. For example, you can use inequalities to determine the maximum or minimum value of a function, or to find the intervals where a function is increasing or decreasing.

Q: What are some advanced topics in inequalities?

A: Some advanced topics in inequalities include:

• Inequalities with multiple variables.
• Inequalities with absolute value.
• Inequalities with quadratic expressions.
• Inequalities with rational expressions.

Q: How can I learn more about inequalities?

A: You can learn more about inequalities by reading textbooks or online resources, working through examples and exercises, and practicing solving inequalities. You can also try taking online courses or attending workshops or conferences on inequalities.