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Chapter 5: Problem 30

For the following problems, solve the inequalities. $$ \frac{y}{7}>3 $$

Short Answer

Expert verified
Question: Solve the inequality $\frac{y}{7} > 3$ and express the solution using interval notation. Answer: $(21, \infty)$

Step by step solution

01

Multiply both sides by 7

To remove the fraction, we will multiply both sides of the inequality by 7. Note that 7 is a positive number, so the inequality sign remains the same: $$ 7 \cdot \frac{y}{7} > 7 \cdot 3 $$
02

Simplify both sides

Now we will cancel out the 7 on the left side and multiply the 7 on the right side by 3: $$ y > 21 $$
03

Write down the solution

The inequality y > 21 is the solution. In interval notation, this can be written as: $$ (y, \infty) $$ So, the solution is all values of y greater than 21.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fractional Inequalities
Fractional inequalities can be intimidating, but once you break them down, they become manageable.
In an inequality involving a fraction, one side of the inequality will generally present as a ratio of two expressions. Our aim is to isolate the variable鈥攈ere, that's \( y \)鈥攂y performing operations that "undo" the fraction.
  • Identify the fraction: In our example, the inequality is \( \frac{y}{7} > 3 \).
  • Eliminate the fraction by multiplying both sides by the denominator, which is 7.
It's important to notice that if the denominator is positive, the direction of the inequality sign doesn't change when we multiply both sides. However, if the denominator were negative, you'd need to reverse the inequality sign. Being mindful of these rules ensures accurate solutions.
Solving Inequalities
Solving inequalities involves many of the same steps as solving equations, with a few additional rules to keep track of.
  • When we solve the inequality \( \frac{y}{7} > 3 \), we multiply both sides by 7, which is a positive number.
  • The inequality you're working with may require you to either add, subtract, multiply, or divide. Each method will aid in isolating the variable.
A crucial step in solving inequalities is remembering how operations affect the inequality sign:- **Multiplying or dividing** by a negative number flips the inequality sign.- **Adding or subtracting** doesn't affect the inequality sign.Ensure to always simplify after each step. As with the example, after multiplying, we obtained \( y > 21 \), which means the variable \( y \) must be greater than 21. This systematic approach will help ensure you solve inequalities correctly.
Interval Notation
Once you've solved the inequality, it's important to express your answer clearly. Interval notation offers a concise way to do just that.Our solution \( y > 21 \) implies that \( y \) can take any value greater than 21 but not 21 itself.
  • In interval notation, this is written as \((21, \infty)\).
  • Round brackets \(()\) indicate that the endpoint is not included, which corresponds to an open interval in this context.
This makes it straightforward for anyone reviewing your solution to understand the range of values that satisfy the inequality. You'll often see solutions written this way in mathematics because it is a standardized method of representing the set of possible solutions.

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