Solve The Inequality:$\frac{x}{-3} \leqslant -6$

Introduction to Inequalities

Inequalities are mathematical expressions that compare two values or expressions, indicating whether one is greater than, less than, greater than or equal to, or less than or equal to the other. Inequalities are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving the inequality $\frac{x}{-3} \leqslant -6$.

Understanding the Inequality

The given inequality is $\frac{x}{-3} \leqslant -6$. To solve this inequality, we need to isolate the variable $x$. The first step is to multiply both sides of the inequality by $-3$. However, when we multiply or divide both sides of an inequality by a negative number, we need to reverse the direction of the inequality sign.

Multiplying Both Sides by -3

When we multiply both sides of the inequality by $-3$, we get:

$$x \geqslant -18$$

However, we need to remember that we reversed the direction of the inequality sign because we multiplied by a negative number.

Understanding the Solution

The solution to the inequality $x \geqslant -18$ means that $x$ can take any value greater than or equal to $-18$. In other words, $x$ can be $-18$, $-17$, $-16$, and so on, or any positive value.

Graphing the Solution

To visualize the solution, we can graph the inequality on a number line. The number line represents all possible values of $x$. We can plot a point at $-18$ and shade the region to the right of this point, indicating that $x$ can take any value greater than or equal to $-18$.

Solving Inequalities with Variables on Both Sides

In some cases, we may have an inequality with variables on both sides. For example, consider the inequality $2x + 5 \leqslant 3x - 2$. To solve this inequality, we need to isolate the variable $x$.

Subtracting 2x from Both Sides

The first step is to subtract $2x$ from both sides of the inequality:

$$5 \leqslant x - 2$$

Adding 2 to Both Sides

Next, we add $2$ to both sides of the inequality:

$$7 \leqslant x$$

Understanding the Solution

The solution to the inequality $7 \leqslant x$ means that $x$ can take any value greater than or equal to $7$. In other words, $x$ can be $7$, $8$, $9$, and so on, or any positive value.

Graphing the Solution

To visualize the solution, we can graph the inequality on a number line. The number line represents all possible values of $x$. We can plot a point at $7$ and shade the region to the right of this point, indicating that $x$ can take any value greater than or equal to $7$.

Solving Inequalities with Absolute Values

In some cases, we may have an inequality with absolute values. For example, consider the inequality $|x| \leqslant 3$. To solve this inequality, we need to consider two cases: $x \geqslant 0$ and $x < 0$.

Case 1: x ≥ 0

When $x \geqslant 0$, the absolute value of $x$ is equal to $x$. Therefore, we can write the inequality as:

$$x \leqslant 3$$

Case 2: x < 0

When $x < 0$, the absolute value of $x$ is equal to $-x$. Therefore, we can write the inequality as:

$$-x \leqslant 3$$

Multiplying Both Sides by -1

To solve the inequality $-x \leqslant 3$, we multiply both sides by $-1$:

$$x \geqslant -3$$

Understanding the Solution

The solution to the inequality $|x| \leqslant 3$ means that $x$ can take any value between $-3$ and $3$, inclusive. In other words, $x$ can be $-3$, $-2$, $-1$, $0$, $1$, $2$, or $3$.

Graphing the Solution

To visualize the solution, we can graph the inequality on a number line. The number line represents all possible values of $x$. We can plot points at $-3$ and $3$ and shade the region between these points, indicating that $x$ can take any value between $-3$ and $3$, inclusive.

Conclusion

In conclusion, solving inequalities is a crucial skill for students to master. In this article, we have discussed how to solve inequalities with variables on both sides, absolute values, and graphing the solution on a number line. By following these steps and understanding the solution, we can visualize the solution on a number line and represent all possible values of the variable.

Frequently Asked Questions

• Q: What is an inequality? A: An inequality is a mathematical expression that compares two values or expressions, indicating whether one is greater than, less than, greater than or equal to, or less than or equal to the other.
• Q: How do I solve an inequality? A: To solve an inequality, you need to isolate the variable by performing operations on both sides 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 $ax + b \leqslant c$, where $a$, $b$, and $c$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c \leqslant 0$, where $a$, $b$, and $c$ are constants.
• Q: How do I graph an inequality on a number line? A: To graph an inequality on a number line, you need to plot a point at the value of the variable and shade the region to the left or right of the point, depending on the direction of the inequality sign.

References

• [1] "Inequalities" by Khan Academy
• [2] "Solving Inequalities" by Mathway
• [3] "Graphing Inequalities" by Purplemath

Note: The references provided are for educational purposes only and are not intended to be a comprehensive list of resources.

Introduction

Solving inequalities can be a challenging task, but with the right guidance, it can become a breeze. In this article, we will provide a comprehensive Q&A guide to help you understand and solve inequalities.

Q: What is an inequality?

A: An inequality is a mathematical expression that compares two values or expressions, indicating whether one is greater than, less than, greater than or equal to, or less than or equal to the other.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable by performing operations on both sides of the inequality. This may involve adding, subtracting, multiplying, or dividing both sides of the inequality by a constant.

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

Q: How do I graph an inequality on a number line?

A: To graph an inequality on a number line, you need to plot a point at the value of the variable and shade the region to the left or right of the point, depending on the direction of the inequality sign.

Q: What is the order of operations when solving an inequality?

A: The order of operations when solving an inequality is the same as when solving an equation: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right).

Q: Can I add or subtract the same value to both sides of an inequality?

A: Yes, you can add or subtract the same value to both sides of an inequality. This is a valid operation when solving an inequality.

Q: Can I multiply or divide both sides of an inequality by a negative number?

A: No, you cannot multiply or divide both sides of an inequality by a negative number. When you do, you need to reverse the direction of the inequality sign.

Q: How do I solve an inequality with absolute values?

A: To solve an inequality with absolute values, you need to consider two cases: the case when the variable is positive and the case when the variable is negative.

Q: What is the solution to an inequality?

A: The solution to an inequality is the set of all values that satisfy the inequality.

Q: How do I write the solution to an inequality in interval notation?

A: To write the solution to an inequality in interval notation, you need to use the following notation:

• $(-\infty, a)$ to represent all values less than $a$
• $(a, \infty)$ to represent all values greater than $a$
• $[a, b]$ to represent all values between $a$ and $b$, inclusive
• $(a, b)$ to represent all values between $a$ and $b$, exclusive

Q: Can I use a calculator to solve an inequality?

A: Yes, you can use a calculator to solve an inequality. However, you need to be careful when using a calculator to solve an inequality, as it may not always give you the correct solution.

Q: How do I check my solution to an inequality?

A: To check your solution to an inequality, you need to plug the solution back into the original inequality and verify that it is true.

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

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

• Not reversing the direction of the inequality sign when multiplying or dividing both sides by a negative number
• Not considering both cases when solving an inequality with absolute values
• Not checking the solution to an inequality

Conclusion

Solving inequalities can be a challenging task, but with the right guidance, it can become a breeze. By following the steps outlined in this article and avoiding common mistakes, you can become proficient in solving inequalities and apply them to real-world problems.

Frequently Asked Questions

• Q: What is an inequality? A: An inequality is a mathematical expression that compares two values or expressions, indicating whether one is greater than, less than, greater than or equal to, or less than or equal to the other.
• Q: How do I solve an inequality? A: To solve an inequality, you need to isolate the variable by performing operations on both sides 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 $ax + b \leqslant c$, where $a$, $b$, and $c$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c \leqslant 0$, where $a$, $b$, and $c$ are constants.

References

• [1] "Inequalities" by Khan Academy
• [2] "Solving Inequalities" by Mathway
• [3] "Graphing Inequalities" by Purplemath