Solve The Inequality:${ -1 + 6x \leq -1 }$ Or ${ 4 - 9x \leq -59 }$

In mathematics, inequalities are a fundamental concept that plays a crucial role in solving various problems. In this article, we will focus on solving two given inequalities: $-1 + 6x \leq -1$ and $4 - 9x \leq -59$. We will break down each inequality, explain the steps involved, and provide a clear solution.

Understanding Inequalities

An inequality is a statement that compares two expressions using a mathematical symbol, such as less than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥). Inequalities can be linear, quadratic, or even more complex, depending on the expressions involved.

Solving the First Inequality: $-1 + 6x \leq -1$

To solve the first inequality, we need to isolate the variable $x$. We can start by adding $1$ to both sides of the inequality:

$$-1 + 6x \leq -1$$

$$6x \leq 0$$

Next, we can divide both sides of the inequality by $6$:

$$x \leq 0$$

This means that the solution to the first inequality is $x \leq 0$.

Solving the Second Inequality: $4 - 9x \leq -59$

To solve the second inequality, we need to isolate the variable $x$. We can start by subtracting $4$ from both sides of the inequality:

$$4 - 9x \leq -59$$

$$-9x \leq -63$$

Next, we can divide both sides of the inequality by $-9$. However, when we divide or multiply an inequality by a negative number, we need to reverse the direction of the inequality sign:

$$x \geq 7$$

This means that the solution to the second inequality is $x \geq 7$.

Combining the Solutions

Since we have two inequalities, we need to find the intersection of their solutions. In this case, the first inequality has a solution of $x \leq 0$, while the second inequality has a solution of $x \geq 7$. To find the intersection, we need to find the values of $x$ that satisfy both inequalities.

The intersection of the two solutions is the range of values that satisfy both inequalities. In this case, the intersection is the range of values that satisfy both $x \leq 0$ and $x \geq 7$. However, since these two inequalities are mutually exclusive, the intersection is empty.

Conclusion

In this article, we solved two given inequalities: $-1 + 6x \leq -1$ and $4 - 9x \leq -59$. We broke down each inequality, explained the steps involved, and provided a clear solution. We also combined the solutions to find the intersection of the two inequalities. The intersection is empty, indicating that there are no values of $x$ that satisfy both inequalities.

Tips and Tricks

When solving inequalities, it's essential to follow the order of operations and to isolate the variable on one side of the inequality. Additionally, when dividing or multiplying an inequality by a negative number, we need to reverse the direction of the inequality sign.

Common Mistakes

When solving inequalities, some common mistakes include:

• Not following the order of operations
• Not isolating the variable on one side of the inequality
• Not reversing the direction of the inequality sign when dividing or multiplying by a negative number

Real-World Applications

Inequalities have numerous real-world applications, including:

• Finance: Inequalities are used to model financial transactions, such as investments and loans.
• Science: Inequalities are used to model physical phenomena, such as motion and energy.
• Engineering: Inequalities are used to design and optimize systems, such as bridges and buildings.

Conclusion

In our previous article, we explored the concept of solving inequalities and provided a step-by-step guide to solving two given inequalities. In this article, we will address some common questions and concerns that students and educators may have when it comes to solving inequalities.

Q: What is the difference between an inequality and an equation?

A: An equation is a statement that says two expressions are equal, while an inequality is a statement that says one expression is less than, greater than, less than or equal to, or greater than or equal to another expression.

Q: How do I know which direction to flip the inequality sign when multiplying or dividing by a negative number?

A: When multiplying or dividing an inequality by a negative number, you need to flip the direction of the inequality sign. For example, if you have the inequality $x > 5$ and you multiply both sides by $-2$, the inequality becomes $-2x < -10$. Note that the direction of the inequality sign has been flipped.

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. For example, if you have the inequality $x > 5$ and you add $3$ to both sides, the inequality becomes $x + 3 > 8$.

Q: Can I multiply or divide both sides of an inequality by the same value?

A: Yes, you can multiply or divide both sides of an inequality by the same value, but be careful not to flip the direction of the inequality sign if the value is negative.

Q: How do I solve an inequality with a variable on both sides?

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

Q: Can I use the same steps to solve a system of inequalities as I would to solve a system of equations?

A: No, you cannot use the same steps to solve a system of inequalities as you would to solve a system of equations. When solving a system of inequalities, you need to find the intersection of the solutions to each inequality.

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 on the number line that represents the solution to the inequality. If the inequality is of the form $x > a$, you would plot a point to the right of $a$. If the inequality is of the form $x < a$, you would plot a point to the left of $a$.

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

A: Yes, you can use a calculator to solve an inequality, but be careful not to enter the wrong values or operations. It's always a good idea to double-check your work and make sure that the solution makes sense.

Conclusion

In conclusion, solving inequalities can be a challenging but rewarding experience. By following the steps outlined in this article and practicing with different types of inequalities, you can become more confident and proficient in solving inequalities. Remember to always follow the order of operations, isolate the variable on one side of the inequality, and reverse the direction of the inequality sign when dividing or multiplying by a negative number.

Tips and Tricks

• When solving inequalities, it's essential to follow the order of operations and to isolate the variable on one side of the inequality.
• When dividing or multiplying an inequality by a negative number, you need to reverse the direction of the inequality sign.
• When solving a system of inequalities, you need to find the intersection of the solutions to each inequality.
• When graphing an inequality on a number line, you need to plot a point on the number line that represents the solution to the inequality.

Common Mistakes

• Not following the order of operations
• Not isolating the variable on one side of the inequality
• Not reversing the direction of the inequality sign when dividing or multiplying by a negative number
• Not finding the intersection of the solutions to each inequality when solving a system of inequalities

Real-World Applications

Inequalities have numerous real-world applications, including:

• Finance: Inequalities are used to model financial transactions, such as investments and loans.
• Science: Inequalities are used to model physical phenomena, such as motion and energy.
• Engineering: Inequalities are used to design and optimize systems, such as bridges and buildings.

Conclusion

In conclusion, solving inequalities is a crucial skill in mathematics that has numerous real-world applications. By following the steps outlined in this article and practicing with different types of inequalities, you can become more confident and proficient in solving inequalities. Remember to always follow the order of operations, isolate the variable on one side of the inequality, and reverse the direction of the inequality sign when dividing or multiplying by a negative number.