Writing the Solution in Interval Notation

Oct 13, 2025 by miodeya.com 22 views

=====================================================

Inequalities are mathematical expressions that compare two values, indicating whether one is greater than, less than, or equal to the other. Solving inequalities involves isolating the variable on one side of the inequality sign. In this article, we will walk through a step-by-step guide on solving inequalities, using the given problem as an example.

Step 1: Simplifying the Inequality

The given inequality is ${$ -10 + 8x \ \textless \ 6x - 4$}$. To simplify this inequality, we need to combine like terms and isolate the variable.

Combining Like Terms

First, let's combine the like terms on both sides of the inequality.

${$ -10 + 8x \ \textless \ 6x - 4$}$

Subtracting ${ $6x}$ from both sides gives us:

${$ -10 + 2x \ \textless \ -4$}$

Adding ${ $10}$ to both sides gives us:

${ $2x \ \textless \ 6\$}$

Isolating the Variable

Now, let's isolate the variable by dividing both sides of the inequality by ${ $2}$ .

${ $2x \ \textless \ 6\$}$

Dividing both sides by ${ $2}$ gives us:

${$ x \ \textless \ 3$}$

Step 2: Solving the Inequality

The given inequality is ${$ -10 \ \textless \ -2x - 4$}$. To solve this inequality, we need to isolate the variable.

Adding 4 to Both Sides

First, let's add ${ $4}$ to both sides of the inequality.

${$ -10 \ \textless \ -2x - 4$}$

Adding ${ $4}$ to both sides gives us:

${$ -6 \ \textless \ -2x$}$

Dividing Both Sides by -2

Now, let's divide both sides of the inequality by ${$ -2}$. Remember to reverse the inequality sign when dividing by a negative number.

${$ -6 \ \textless \ -2x$}$

Dividing both sides by ${$ -2}$ gives us:

${ $3 \ \textgreater \ x\$}$

Step 3: Checking the Solution

The given inequality is ${$ -6 \ \textless \ -2x$}$. To check the solution, we need to plug in the value of ${$ x}$ into the original inequality.

Plugging in x = 3

Let's plug in ${$ x = 3}$ into the original inequality.

${$ -6 \ \textless \ -2(3)$}$

Simplifying the expression gives us:

${$ -6 \ \textless \ -6$}$

This is a true statement, so the solution ${$ x = 3}$ is valid.

Step 4: Finalizing the Solution

The final step in solving the inequality is to write the solution in interval notation.

Writing the Solution in Interval Notation

The solution to the inequality is ${$ x \ \textgreater \ 3$}$. In interval notation, this can be written as:

${$ (3, \infty)$}$

This means that the solution is all real numbers greater than ${ $3}$ .

Conclusion

Solving inequalities involves isolating the variable on one side of the inequality sign. By following the steps outlined in this article, we can solve inequalities and write the solution in interval notation. Remember to check the solution by plugging in the value of the variable into the original inequality. With practice, you will become proficient in solving inequalities and be able to apply this skill to a wide range of mathematical problems.

Frequently Asked Questions

What is the final step in solving the inequality?
- The final step in solving the inequality is to write the solution in interval notation.
How do I check the solution?
- To check the solution, plug in the value of the variable into the original inequality.
What is the solution to the inequality?
- The solution to the inequality is ${$ x \ \textgreater \ 3$}$.

Key Takeaways

Solving inequalities involves isolating the variable on one side of the inequality sign.
To check the solution, plug in the value of the variable into the original inequality.
The solution to the inequality is ${$ x \ \textgreater \ 3$}$.

Q: What is the first step in solving an inequality?

A: The first step in solving an inequality is to simplify the expression by combining like terms and isolating the variable.

Q: How do I know which direction to move the inequality sign when adding or subtracting the same value to both sides?

A: When adding or subtracting the same value to both sides of an inequality, the direction of the inequality sign remains 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 \ \textless \ cx + d$}$, where ${$ a}$, ${$ b}$, ${$ c}$, and ${$ d}$ are constants. A quadratic inequality is an inequality that can be written in the form ${$ ax^2 + bx + c \ \textless \ dx^2 + ex + f$}$, where ${$ a}$, ${$ b}$, ${$ c}$, ${$ d}$, ${$ e}$, and ${$ f}$ 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 final step in solving an inequality?

A: The final step in solving an inequality is to write the solution in interval notation.

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

A: To check the solution to an inequality, plug in the value of the variable into the original inequality.

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

A: A strict inequality is an inequality that uses the symbols ${$ \lt$}$ or ${$ \gt$}$, while a non-strict inequality is an inequality that uses the symbols ${$ \leq$}$ or ${$ \geq$}$.

Q: How do I solve a system of inequalities?

A: To solve a system of inequalities, you can use the following steps:

Graph each inequality on a coordinate plane.
Find the intersection of the two graphs.
Write the solution in interval notation.

Q: What is the importance of solving inequalities?

A: Solving inequalities is an important skill in mathematics, as it allows you to compare two values and determine which one is greater or less than the other. Inequalities are used in a wide range of applications, including finance, science, and engineering.

Q: How can I practice solving inequalities?

A: You can practice solving inequalities by working through example problems, using online resources, or taking a math course.

Conclusion

Solving inequalities is an important skill in mathematics, and it requires practice and patience to master. By following the steps outlined in this article, you can solve inequalities and write the solution in interval notation. Remember to check the solution by plugging in the value of the variable into the original inequality. With practice, you will become proficient in solving inequalities and be able to apply this skill to a wide range of mathematical problems.

Frequently Asked Questions

What is the first step in solving an inequality?
- The first step in solving an inequality is to simplify the expression by combining like terms and isolating the variable.
How do I know which direction to move the inequality sign when adding or subtracting the same value to both sides?
- When adding or subtracting the same value to both sides of an inequality, the direction of the inequality sign remains the same.
What is the difference between a linear inequality and a quadratic inequality?
- A linear inequality is an inequality that can be written in the form ${$ ax + b \ \textless \ cx + d$}$, while a quadratic inequality is an inequality that can be written in the form ${$ ax^2 + bx + c \ \textless \ dx^2 + ex + f$}$.

Key Takeaways

Solving inequalities involves simplifying the expression, isolating the variable, and writing the solution in interval notation.
To check the solution to an inequality, plug in the value of the variable into the original inequality.
The final step in solving an inequality is to write the solution in interval notation.