Solve The Following Inequality. Express The Solution Using Interval Notation.$\frac{x+4}{7} \leq \frac{x+9}{5}-\frac{13}{35}$The Solution Set Is $\square$Hint: Multiply Each Term By The Least Common Multiple Of The Denominators:

Introduction

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 a specific type of inequality, namely the rational inequality, and express the solution using interval notation. We will use the given inequality $\frac{x+4}{7} \leq \frac{x+9}{5}-\frac{13}{35}$ as an example and provide a step-by-step guide on how to solve it.

Understanding the Inequality

Before we dive into solving the inequality, let's first understand what it means. The given inequality is a rational inequality, which means it involves fractions or rational expressions. The inequality states that the expression $\frac{x+4}{7}$ is less than or equal to the expression $\frac{x+9}{5}-\frac{13}{35}$. Our goal is to find the values of $x$ that satisfy this inequality.

Multiplying Each Term by the Least Common Multiple of the Denominators

The hint provided in the problem statement suggests that we multiply each term by the least common multiple (LCM) of the denominators. In this case, the denominators are 7, 5, and 35. The LCM of these numbers is 35. By multiplying each term by 35, we can eliminate the fractions and simplify the inequality.

\frac{35(x+4)}{7} \leq \frac{35(x+9)}{5} - \frac{35(13)}{35}

Simplifying the Inequality

Now that we have eliminated the fractions, we can simplify the inequality by multiplying out the numerators and denominators.

5(x+4) \leq 7(x+9) - 13

Expanding and Simplifying the Inequality

Next, we need to expand and simplify the inequality by distributing the coefficients to the terms inside the parentheses.

5x + 20 \leq 7x + 63 - 13

Combining Like Terms

Now, we can combine like terms on both sides of the inequality.

5x + 20 \leq 7x + 50

Subtracting 5x from Both Sides

To isolate the variable $x$, we need to subtract 5x from both sides of the inequality.

20 \leq 2x + 50

Subtracting 50 from Both Sides

Next, we need to subtract 50 from both sides of the inequality.

-30 \leq 2x

Dividing Both Sides by 2

Finally, we can divide both sides of the inequality by 2 to solve for $x$.

-15 \leq x

Expressing the Solution in Interval Notation

The solution to the inequality is $x \leq -15$. In interval notation, this can be expressed as $(-\infty, -15]$.

Conclusion

In this article, we have solved the rational inequality $\frac{x+4}{7} \leq \frac{x+9}{5}-\frac{13}{35}$ and expressed the solution using interval notation. We used the hint provided in the problem statement to multiply each term by the least common multiple of the denominators, which allowed us to eliminate the fractions and simplify the inequality. By following the steps outlined in this article, students can develop the skills necessary to solve rational inequalities and express their solutions in interval notation.

Frequently Asked Questions

• Q: What is the least common multiple (LCM) of the denominators 7, 5, and 35? A: The LCM of 7, 5, and 35 is 35.
• Q: Why do we need to multiply each term by the LCM of the denominators? A: We need to multiply each term by the LCM of the denominators to eliminate the fractions and simplify the inequality.
• Q: How do we express the solution to the inequality in interval notation? A: The solution to the inequality can be expressed in interval notation as $(-\infty, -15]$.

Additional Resources

• For more information on solving rational inequalities, see the article "Solving Rational Inequalities: A Step-by-Step Guide".
• For more information on interval notation, see the article "Interval Notation: A Guide to Expressing Solutions to Inequalities".

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "College Algebra" by James Stewart
• [3] "Intermediate Algebra" by Charles P. McKeague

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