Problem 7 - Algebra Form and Function

Question

Each of the inequalities can be solved by performing a single operation on both sides. State the operation, indicating whether or not the inequality changes direction. Solve the inequality. $$ 12 x \geq 60 $$

Step-by-Step Solution

Answer

Question: Solve the inequality \(12x \geq 60\) and provide the solution. Answer: The solution to the inequality is \(x \geq 5\).

Step 1: Identify the operation

To solve the inequality \(12x \geq 60\), we need to find the value of \(x\) that satisfies this equation. The operation that we need to perform is division. We will divide both sides of the inequality by \(12\).

Step 2: Determine if the inequality changes direction

Since we are dividing both sides of the inequality by a positive number (\(12\)), the inequality will not change direction. This means that the inequality will remain in its current form, \(\geq\).

Step 3: Solve the inequality

Now, we will divide both sides of the inequality by \(12\): $$ \frac{12x}{12} \geq \frac{60}{12} $$ This simplifies to: $$ x \geq 5 $$ So, the solution to the inequality is \(x \geq 5\).

Key Concepts

Division in Inequalities

When solving inequalities, one of the common operations used is division. This process is very similar to solving regular equations, but with inequalities, you need to pay attention to particular rules. In the case of the inequality \(12x \geq 60\), we aim to isolate \(x\) on one side of the inequality. To do this, we divide both sides by the coefficient of \(x\), which is \(12\).

Division allows us to simplify the inequality and find the value that \(x\) must satisfy. This step is crucial as it directly transforms the inequality into a more understandable form such as \(x \geq 5\). Here are the steps involved:

• Identify the number by which both sides of the inequality must be divided.
• Ensure you are dividing by a positive number to maintain the direction of the inequality.
• Perform the division operation on both sides.

This method efficiently simplifies terms and helps in direct problem-solving with clear results. Remember, division serves as a useful tool in moving towards the solution of any linear inequality.

Inequality Direction Change

A unique aspect of solving inequalities compared to solving equations is the possible change in direction of the inequality sign. While dividing both sides of an inequality, it is essential to check whether the direction of the inequality sign needs to change. This only happens when dividing or multiplying both sides by a negative number.

For instance, if we had to divide by \(-12\) instead of \(12\), the inequality sign would flip. However, in our example, we are dividing by \(12\), which is positive. Therefore, the inequality sign \(\geq\) remains unchanged.

It's important to remember:

• Division by a positive number keeps the inequality direction the same.
• Division by a negative number reverses the inequality direction.

This rule is crucial and ensures accuracy in solving inequalities. Being aware of this principle helps prevent common mistakes where the inequality direction is mistakenly altered, leading to incorrect solutions.

Linear Inequalities

Linear inequalities represent expressions related by inequality signs such as greater than, less than, greater than or equal to, or less than or equal to. In our example, \(12x \geq 60\), we have a linear inequality because it contains a linear expression of \(x\).

Solving linear inequalities follows the same general methods as solving linear equations but includes special attention to the rules for inequalities, especially with regards to operations that can affect the inequality direction. In solving the inequality:

• First, identify the operation needed to isolate the variable.
• Next, apply necessary operations (such as addition, subtraction, multiplication, or division).
• Finally, check whether the operations change the inequality's direction.

The solution \(x \geq 5\) indicates that \(x\) can be any number greater than or equal to 5. This is a simple form of a linear inequality, demonstrating how basic algebraic techniques can be applied to find the range of values that satisfy the inequality.