Problem 5 - Prealgebra

Question

Solve each inequality. Check your solution. $$\frac{3}{4} \leq \frac{5}{7} y$$

Step-by-Step Solution

Answer

The solution is \( y \geq \frac{21}{20} \).

Step 1: Understand the Inequality

We need to solve the inequality \( \frac{3}{4} \leq \frac{5}{7} y \) for \( y \). This means finding the values of \( y \) that satisfy the given inequality.

Step 2: Isolate the Variable

To isolate \( y \), multiply each side of the inequality by the reciprocal of \( \frac{5}{7} \), which is \( \frac{7}{5} \). Thus, the inequality becomes: \[ \frac{7}{5} \times \frac{3}{4} \leq y \].

Step 3: Simplify the Left Side

Now simplify the left side: \[ \frac{7}{5} \times \frac{3}{4} = \frac{7 \times 3}{5 \times 4} = \frac{21}{20} \].

Step 4: Rewrite the Inequality

Now, we can rewrite the inequality as \( \frac{21}{20} \leq y \). This means that \( y \) must be greater than or equal to \( \frac{21}{20} \).

Step 5: Check the Solution

To verify, assume \( y = \frac{21}{20} \) and substitute it back into the original inequality: \( \frac{5}{7} \times \frac{21}{20} \). This simplifies to \( \frac{3}{4} \), confirming that the inequality holds as \( \frac{3}{4} \leq \frac{3}{4} \). Thus, \( y \geq \frac{21}{20} \) is correct.

Key Concepts

Understanding Inequalities

In mathematics, inequalities express a relationship where one value is less than, greater than or equal to another. In the exercise:

• The inequality is represented as \( \frac{3}{4} \leq \frac{5}{7} y \), meaning we are looking for values of \( y \) that make the left side less than or equal to the right side.
• This inequality includes a fraction and a variable, indicating a more complex relationship than a simple numerical comparison.

Breaking it down, understanding inequalities involves recognizing the symbols like \( \leq \) which stands for "less than or equal to." Concepts like these allow us to apply mathematical manipulation to find ranges of valid numbers rather than exact values, adding a layer of depth to our solutions.Understanding inequalities is crucial because it helps us solve problems where the exact solution might not be possible and instead, we define a set of possible solutions.

Isolating Variables

To solve any inequality, a common strategy is to isolate the variable on one side. In this problem, we want to solve for \( y \), which currently is tied to a fraction. The steps can be simplified as follows:

• Identify the coefficient of the variable—in this case, \( \frac{5}{7} \).
• Multiply both sides of the inequality by the reciprocal of this coefficient. The reciprocal of \( \frac{5}{7} \) is \( \frac{7}{5} \), which essentially cancels out the fraction when applied to \( y \).

This action maintains the balance of the inequality while simplifying the expression. Once multiplied, we simplify the expression on the left side:
\[ \frac{7}{5} \times \frac{3}{4} = \frac{21}{20} \].Now the inequality \( \frac{21}{20} \leq y \) makes it clear that \( y \) must be greater than or equal to \( \frac{21}{20} \). This process is crucial in problem-solving as it systematically leads you to manage the terms around the variable, allowing for a simpler path to the solution.

Checking Solutions

The final step in solving inequalities is to verify that your solution works within the original problem's framework. It’s important because it confirms that the manipulation of the inequality has preserved its truth. Here's how you do it:

• Substitute the boundary value back into the original inequality.
• If using \( y = \frac{21}{20} \), substitute into \( \frac{5}{7} y \) to see: \( \frac{5}{7} \times \frac{21}{20} \).

Simplifying this returns: \[ \frac{5 \times 21}{7 \times 20} = \frac{105}{140} = \frac{3}{4} \],which matches the left side of our original inequality, confirming that the solution holds.
The solution check is crucial as it reassures us that all steps are followed correctly and no errors were made in calculation. This practice not only reinforces comprehension of the problem but also assures accuracy in mathematical reasoning.