Problem 4 - Prealgebra

Question

Solve each inequality. Check your solution. $$\frac{7}{2} y>63$$

Step-by-Step Solution

Answer

The solution is \( y > 18 \).

Step 1: Understand the Inequality

We are given the inequality \( \frac{7}{2} y > 63 \). This means that \( \frac{7}{2} imes y \) is greater than 63. Our task is to solve for \( y \).

Step 2: Isolate the Variable

To isolate \( y \), we need to remove the fraction \( \frac{7}{2} \) from the left side of the inequality. To do this, multiply both sides of the inequality by the reciprocal of \( \frac{7}{2} \), which is \( \frac{2}{7} \). This gives us: \[ \left( \frac{2}{7} \right) \times \left( \frac{7}{2} y \right) > 63 \times \frac{2}{7} \] Simplifying the left side, the \( \frac{7}{2} \) and \( \frac{2}{7} \) cancel each other out, leaving: \[ y > \frac{126}{7} \]

Step 3: Simplify the Expression

Next, simplify \( \frac{126}{7} \). Divide 126 by 7 to find: \[ \frac{126}{7} = 18 \] So, \( y > 18 \).

Step 4: Solution Verification

To verify, we will test a number greater than 18 to see if it satisfies the original inequality. Choose \( y = 19 \), then calculate: \[ \frac{7}{2} imes 19 = 66.5 \] Since \( 66.5 > 63 \), the solution holds. We can also check a number less than or equal to 18, such as 18. \( \frac{7}{2} \times 18 = 63 \), which does not satisfy the inequality because 63 is not greater than 63.

Key Concepts

Prealgebra

Prealgebra is a vital foundation for understanding higher-level math concepts. It serves as the bridge from basic arithmetic to the introduction of algebraic reasoning. In prealgebra, you begin to see numbers not just as standalone digits but as part of expressions or equations. This subject introduces variables, which are symbols (often letters) that represent unknown values.

• This stage helps students grasp mathematical properties and operations.
• Establishes the basic understanding necessary to solve equations and inequalities.
• Supports logical thinking and problem-solving skills.

In our inequality exercise, understanding the arithmetic operations and algebraic expressions lays the groundwork for effectively solving the inequality. It's crucial to master these prealgebra basics to tackle more complex problems successfully.

Solving Inequalities

Solving inequalities is similar to solving equations, however, it involves expressions using inequality signs such as '>', '<', '≥', or '≤'. These symbols indicate the relationship between two expressions.
When solving inequalities, the goal is to find the range of values that make the inequality true. The key steps include:

• Identifying and understanding the inequality.
• Isolating the variable on one side of the inequality.
• Simplifying the resulting expressions.

One core aspect of solving inequalities is ensuring that when you multiply or divide an inequality by a negative number, the inequality sign flips. Although this was not necessary in our exercise with \( \frac{7}{2} y > 63 \), it's essential to remember for future problems. Mastering these techniques will ease problem-solving and foster a better understanding of more complex mathematical concepts.

Step-by-Step Solution

Working through a problem step-by-step provides structure and clarity. Here's a closer look at the essential steps to solve the given inequality:

• **Understand the problem:** The inequality \( \frac{7}{2} y > 63 \) means our expression on the left should be greater than 63.
• **Isolate the variable:** Multiply both sides by the reciprocal of \( \frac{7}{2} \), which is \( \frac{2}{7} \), to get \( y > \frac{126}{7} \).
• **Simplify the expression:** Calculating \( \frac{126}{7} \) results in 18, so \( y > 18 \).

Breaking down the problem into small, manageable parts allows for an easier comprehension of each step. It is crucial not just to find the final solution but also to understand how each step contributes to the answer.

Verify Solution

Verifying the solution to an inequality is a crucial part of the problem-solving process. After solving, we must ensure our solution is correct and satisfies the original inequality.

• **Choose a test value:** Pick any value for \( y \) that exceeds the solution, e.g., \( y = 19 \).
• **Perform substitution:** Substitute 19 back into the original inequality. Calculate \( \frac{7}{2} \times 19 \) to check if it is > 63.
• **Evaluate the results:** Since the result 66.5 > 63, it confirms the solution is correct.

Verification helps strengthen understanding by confirming that the variable's values indeed satisfy the inequality. Regularly practicing this step ensures mathematical accuracy and boosts confidence in handling inequalities.