Solve The Inequality:$\[ -9(2x + 1) \geq 36 \\]

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Introduction

Inequalities are mathematical expressions that compare two values, and they are used to describe relationships between variables. Solving inequalities is an essential skill in mathematics, and it requires a deep understanding of algebraic concepts. In this article, we will focus on solving the inequality $-9(2x + 1) \geq 36$, and we will provide a step-by-step guide to help you understand the solution.

Understanding the Inequality

The given inequality is $-9(2x + 1) \geq 36$. To solve this inequality, we need to isolate the variable $x$ on one side of the inequality sign. The inequality sign $\geq$ means that the expression on the left-hand side is greater than or equal to the expression on the right-hand side.

Step 1: Distribute the Negative 9

The first step in solving the inequality is to distribute the negative 9 to the terms inside the parentheses. This will give us:

$$-18x - 9 \geq 36$$

Step 2: Add 9 to Both Sides

Next, we need to add 9 to both sides of the inequality to get rid of the negative term. This will give us:

$$-18x \geq 45$$

Step 3: Divide Both Sides by -18

Now, we need to divide both sides of the inequality by -18. However, when we divide or multiply an inequality by a negative number, we need to reverse the direction of the inequality sign. This will give us:

$$x \leq -\frac{45}{18}$$

Step 4: Simplify the Fraction

The fraction $-\frac{45}{18}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 9. This will give us:

$$x \leq -\frac{5}{2}$$

Conclusion

In conclusion, the solution to the inequality $-9(2x + 1) \geq 36$ is $x \leq -\frac{5}{2}$. This means that any value of $x$ that is less than or equal to $-\frac{5}{2}$ will satisfy the inequality.

Understanding the Solution

To understand the solution, let's consider a few examples. If we substitute $x = -3$ into the inequality, we get:

$$-9(2(-3) + 1) \geq 36$$

Simplifying this expression, we get:

$$-9(-5) \geq 36$$

$$45 \geq 36$$

This is true, so $x = -3$ is a solution to the inequality. On the other hand, if we substitute $x = -2$ into the inequality, we get:

$$-9(2(-2) + 1) \geq 36$$

Simplifying this expression, we get:

$$-9(-3) \geq 36$$

$$27 \geq 36$$

This is false, so $x = -2$ is not a solution to the inequality.

Graphing the Solution

To visualize the solution, we can graph the inequality on a number line. The number line is a line that extends infinitely in both directions, and it is used to represent the values of a variable. To graph the inequality, we need to plot a point on the number line that represents the solution to the inequality. In this case, the solution is $x \leq -\frac{5}{2}$, so we can plot a point at $x = -\frac{5}{2}$ on the number line.

Conclusion

In conclusion, solving inequalities is an essential skill in mathematics, and it requires a deep understanding of algebraic concepts. In this article, we have focused on solving the inequality $-9(2x + 1) \geq 36$, and we have provided a step-by-step guide to help you understand the solution. We have also discussed the importance of understanding the solution and graphing the solution on a number line.

Frequently Asked Questions

• Q: What is the solution to the inequality $-9(2x + 1) \geq 36$? A: The solution to the inequality is $x \leq -\frac{5}{2}$.
• Q: How do I graph the solution on a number line? A: To graph the solution on a number line, you need to plot a point at $x = -\frac{5}{2}$.
• Q: What is the importance of understanding the solution to an inequality? A: Understanding the solution to an inequality is essential in mathematics, as it allows you to visualize the relationship between the variables and make informed decisions.

Final Thoughts

Solving inequalities is a critical skill in mathematics, and it requires a deep understanding of algebraic concepts. In this article, we have focused on solving the inequality $-9(2x + 1) \geq 36$, and we have provided a step-by-step guide to help you understand the solution. We have also discussed the importance of understanding the solution and graphing the solution on a number line. By following the steps outlined in this article, you will be able to solve inequalities with confidence and accuracy.

Additional Resources

• Khan Academy: Solving Inequalities
• Mathway: Solving Inequalities
• Wolfram Alpha: Solving Inequalities

References

• "Algebra and Trigonometry" by Michael Sullivan
• "College Algebra" by James Stewart
• "Mathematics for the Nonmathematician" by Morris Kline
  

Introduction

Solving inequalities is a critical skill in mathematics, and it requires a deep understanding of algebraic concepts. In our previous article, we provided a step-by-step guide to solving the inequality $-9(2x + 1) \geq 36$. In this article, we will provide a Q&A guide to help you understand the solution and address any questions you may have.

Q&A Guide

Q: What is the solution to the inequality $-9(2x + 1) \geq 36$?

A: The solution to the inequality is $x \leq -\frac{5}{2}$.

Q: How do I graph the solution on a number line?

A: To graph the solution on a number line, you need to plot a point at $x = -\frac{5}{2}$.

Q: What is the importance of understanding the solution to an inequality?

A: Understanding the solution to an inequality is essential in mathematics, as it allows you to visualize the relationship between the variables and make informed decisions.

Q: How do I determine if a value is a solution to an inequality?

A: To determine if a value is a solution to an inequality, you need to substitute the value into the inequality and check if the inequality is true.

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

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to factor the quadratic expression and then use the sign chart method to determine the solution.

Q: What is the sign chart method?

A: The sign chart method is a technique used to determine the solution to a quadratic inequality by creating a chart that shows the sign of the quadratic expression for different values of the variable.

Q: How do I use the sign chart method to solve a quadratic inequality?

A: To use the sign chart method, you need to create a chart that shows the sign of the quadratic expression for different values of the variable. Then, you need to determine the intervals where the quadratic expression is positive or negative.

Q: What is the difference between a rational inequality and a polynomial inequality?

A: A rational inequality is an inequality that can be written in the form $\frac{f(x)}{g(x)} \geq 0$, where $f(x)$ and $g(x)$ are polynomials. A polynomial inequality is an inequality that can be written in the form $f(x) \geq 0$, where $f(x)$ is a polynomial.

Q: How do I solve a rational inequality?

A: To solve a rational inequality, you need to factor the numerator and denominator and then use the sign chart method to determine the solution.

Q: What is the importance of understanding the concept of absolute value?

A: Understanding the concept of absolute value is essential in mathematics, as it allows you to work with inequalities that involve absolute value.

Q: How do I solve an inequality that involves absolute value?

A: To solve an inequality that involves absolute value, you need to isolate the absolute value expression and then use the definition of absolute value to determine the solution.

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

A: A linear programming problem is a problem that involves maximizing or minimizing a linear function subject to linear constraints. A quadratic programming problem is a problem that involves maximizing or minimizing a quadratic function subject to quadratic constraints.

Q: How do I solve a linear programming problem?

A: To solve a linear programming problem, you need to use the simplex method or the graphical method to determine the optimal solution.

Q: What is the simplex method?

A: The simplex method is a technique used to solve linear programming problems by iteratively improving the solution until an optimal solution is reached.

Q: How do I use the simplex method to solve a linear programming problem?

A: To use the simplex method, you need to create a tableau that shows the coefficients of the variables and the objective function. Then, you need to iteratively improve the solution until an optimal solution is reached.

Conclusion

Solving inequalities is a critical skill in mathematics, and it requires a deep understanding of algebraic concepts. In this article, we have provided a Q&A guide to help you understand the solution and address any questions you may have. We have also discussed the importance of understanding the concept of absolute value and the difference between a linear programming problem and a quadratic programming problem.

Frequently Asked Questions

• Q: What is the solution to the inequality $-9(2x + 1) \geq 36$? A: The solution to the inequality is $x \leq -\frac{5}{2}$.
• Q: How do I graph the solution on a number line? A: To graph the solution on a number line, you need to plot a point at $x = -\frac{5}{2}$.
• Q: What is the importance of understanding the solution to an inequality? A: Understanding the solution to an inequality is essential in mathematics, as it allows you to visualize the relationship between the variables and make informed decisions.

Final Thoughts

Solving inequalities is a critical skill in mathematics, and it requires a deep understanding of algebraic concepts. In this article, we have provided a Q&A guide to help you understand the solution and address any questions you may have. We have also discussed the importance of understanding the concept of absolute value and the difference between a linear programming problem and a quadratic programming problem. By following the steps outlined in this article, you will be able to solve inequalities with confidence and accuracy.

Additional Resources

• Khan Academy: Solving Inequalities
• Mathway: Solving Inequalities
• Wolfram Alpha: Solving Inequalities

References

• "Algebra and Trigonometry" by Michael Sullivan
• "College Algebra" by James Stewart
• "Mathematics for the Nonmathematician" by Morris Kline