Incorrectly distributing numbers

Oct 13, 2025 by wbrum.org 103 views

Introduction

Inequalities are mathematical expressions that compare two values, often using greater than or less than symbols. Solving inequalities involves isolating the variable on one side of the inequality sign, while maintaining the relationship between the two sides. In this article, we will focus on solving the inequality $5x - 9 \geq 12(x - 5)$ , which is a common type of linear inequality.

Understanding the Inequality

The given inequality is $5x - 9 \geq 12(x - 5)$ . To solve this inequality, we need to isolate the variable $x$ on one side of the inequality sign. The first step is to simplify the inequality by distributing the numbers outside the parentheses to the terms inside.

Distributing the Numbers

To distribute the numbers outside the parentheses, we multiply each term inside the parentheses by the number outside. In this case, we multiply $12$ by $(x - 5)$ .

${ 5x - 9 \geq 12x - 60 }$

Combining Like Terms

Now that we have distributed the numbers, we can combine like terms on each side of the inequality. Like terms are terms that have the same variable raised to the same power.

Combining Like Terms

On the left side of the inequality, we have $5x$ and $-9$ . On the right side, we have $12x$ and $-60$ . We can combine these terms by adding or subtracting their coefficients.

${ 5x - 9 \geq 12x - 60 }$

Subtracting $5x$ from both sides gives us:

${ -9 \geq 7x - 60 }$

Adding $60$ to both sides gives us:

${ 51 \geq 7x }$

Isolating the Variable

Now that we have combined like terms, we can isolate the variable $x$ on one side of the inequality. To do this, we divide both sides of the inequality by the coefficient of $x$ , which is $7$ .

Dividing Both Sides

Dividing both sides of the inequality by $7$ gives us:

${ \frac{51}{7} \geq x }$

Conclusion

In conclusion, solving the inequality $5x - 9 \geq 12(x - 5)$ involves simplifying the inequality, combining like terms, and isolating the variable $x$ on one side of the inequality sign. By following these steps, we can find the solution to the inequality, which is $\frac{51}{7} \geq x$ .

Tips and Tricks

When solving inequalities, it's essential to maintain the direction of the inequality sign.
When combining like terms, make sure to add or subtract the coefficients correctly.
When isolating the variable, make sure to divide both sides of the inequality by the correct coefficient.

Real-World Applications

Solving inequalities has many real-world applications, including:

Finance: Inequalities are used to compare the value of investments, such as stocks and bonds.
Science: Inequalities are used to model real-world phenomena, such as population growth and chemical reactions.
Engineering: Inequalities are used to design and optimize systems, such as electrical circuits and mechanical systems.

Common Mistakes

Incorrectly distributing numbers: When distributing numbers, make sure to multiply each term inside the parentheses by the number outside.
Incorrectly combining like terms: When combining like terms, make sure to add or subtract the coefficients correctly.
Incorrectly isolating the variable: When isolating the variable, make sure to divide both sides of the inequality by the correct coefficient.

Final Thoughts

Solving inequalities is a crucial skill in mathematics, and it has many real-world applications. By following the steps outlined in this article, you can solve inequalities with confidence. Remember to maintain the direction of the inequality sign, combine like terms correctly, and isolate the variable by dividing both sides of the inequality by the correct coefficient. With practice and patience, you can become proficient in solving inequalities and apply them to real-world problems.

Introduction

In our previous article, we discussed the steps to solve the inequality $5x - 9 \geq 12(x - 5)$ . In this article, we will address some common questions and concerns that students may have when solving inequalities.

Q&A

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

A: Solving an equation involves finding the value of the variable that makes the equation true, whereas solving an inequality involves finding the values of the variable that make the inequality true.

Q: How do I know which direction to distribute the numbers when solving an inequality?

A: When solving an inequality, you should distribute the numbers in the same way that you would when solving an equation. The direction of the inequality sign will remain the same.

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 can use the following steps:

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

Q: What is the difference between a strict inequality and a non-strict inequality?

A: A strict inequality is an inequality that is written with a strict inequality sign, such as $< $ or $> $. A non-strict inequality is an inequality that is written with a non-strict inequality sign, such as $\leq$ or $\geq$ .

Q: How do I know whether to use a strict or non-strict inequality sign?

A: The choice of inequality sign depends on the context of the problem. If the problem states that the inequality is strict, you should use a strict inequality sign. If the problem states that the inequality is non-strict, you should use a non-strict inequality sign.

Q: Can I use algebraic methods to solve inequalities with absolute values?

A: Yes, you can use algebraic methods to solve inequalities with absolute values. To do this, you can use the following steps:

Write the absolute value expression as a piecewise function.
Solve each piece of the function separately.
Combine the solutions to find the final answer.

Q: How do I know whether to use a linear or quadratic function to model a real-world problem?

A: The choice of function depends on the context of the problem. If the problem involves a linear relationship between two variables, you should use a linear function. If the problem involves a quadratic relationship between two variables, you should use a quadratic function.