Problem 22 - Prealgebra

Question

For the given value, state whether each inequality is true or false. $$\frac{y}{3} \geq 2, y=9$$

Step-by-Step Solution

Answer

The inequality is true for \( y = 9 \).

Step 1/Interpret the Inequality

We need to evaluate the inequality \( \frac{y}{3} \geq 2 \) for the given value \( y = 9 \). The inequality asks us to compare \( \frac{y}{3} \) with 2 and determine if it is greater than or equal to 2.

Step 2/Calculate the Left Side

Substitute the given value of \( y = 9 \) into the expression \( \frac{y}{3} \). This yields \( \frac{9}{3} = 3 \).

Step 3/Evaluate the Inequality

Now compare the result from step 2 with 2. We have 3 on the left side and 2 on the right side of the inequality. We check if 3 is greater than or equal to 2, which is true since 3 is greater than 2.

Step 4/State the Conclusion

Since the comparison \( 3 \geq 2 \) holds true, the inequality \( \frac{y}{3} \geq 2 \) is satisfied when \( y = 9 \). Therefore, the inequality is true for the given value of \( y \).

Key Concepts

Solving Inequalities

In mathematics, inequalities are expressions that define the relationship between two values or expressions, indicating whether one is larger, smaller, or equal to the other. Solving inequalities is an essential skill in prealgebra, as it helps in understanding how different quantities compare to each other.
To solve an inequality, follow these steps:

• Identify the inequality symbol, such as \( \geq \), \( \leq \), \( > \), and \( < \).
• Isolate the variable on one side if necessary by performing algebraic operations, similar to solving equations.
• Remember that multiplying or dividing both sides of an inequality by a negative number reverses the inequality symbol.
• Finally, interpret the solution set, which may consist of a range of values satisfying the inequality.

In this example, the inequality \( \frac{y}{3} \geq 2 \) requires comparing the expression \( \frac{y}{3} \) to 2. By substituting a given value for \( y \), you can confirm if the inequality holds true or not.

Substitution Method

The substitution method is a useful tool in solving inequalities and equations alike. This approach involves replacing a variable with a specific value, allowing you to simplify and solve the mathematical expression or inequality.
For instance, in the inequality \( \frac{y}{3} \geq 2 \), we substitute the given value of \( y = 9 \) into the expression. The substitution results in a simpler expression: \( \frac{9}{3} = 3 \).
With this new expression, you can more easily determine the truth of the inequality by comparing the simplified result with the other side of the inequality. This method is especially helpful when verifying if specific values satisfy given inequalities or equations.

Comparing Numbers

Comparing numbers is a crucial aspect of understanding and working with inequalities. It involves determining which of two numbers is greater, smaller, or if they are equal. In our exercise, after substitution, the result is the number 3, which we need to compare to the number 2.
To compare these numbers, you follow a straightforward process:

• Evaluate the simplified expression or result (3 in this example).
• Examine the inequality symbol to know if you're checking for greater than, lesser than, or equality.
• Check if the resulting value satisfies the inequality condition.

For \( 3 \geq 2 \), you observe that 3 is indeed greater than 2, fulfilling the inequality condition. Understanding how to compare numbers in inequalities enables you to effectively communicate which values satisfy certain conditions, providing clarity and confidence in solving these mathematical expressions.