Problem 80 - Elementary Algebra Within Reach

Question

Determine whether the statement is true or false. Justify your answer. The inequality \(-\frac{1}{2} x+6>0\) is equivalent to \(x>12\).

Step-by-Step Solution

Answer

The statement is false.

Step 1: Simplify the Inequality

Isolate \(x\) in the inequality \(-\frac{1}{2} x+6>0\). This involves two steps. First, line up all terms involving \(x\) on one side of the inequality and the constant terms on the other side. This gives \(-\frac{1}{2} x > -6\). Second, to get rid of the negative sign on the left side, multiply both sides by -2 (remember that when you multiply or divide an inequality by a negative number, you must flip the direction of the inequality). Thus, we get \(x < 12\).

Step 2: Compare the Results

Now that we have \(x < 12\) as the simplified form of \(-\frac{1}{2} x+6>0\), we can clearly see that this is not equivalent to \(x > 12\). The direction of the inequality is the opposite.

Step 3: Final Answer

Since \(x < 12\) is not equivalent to \(x > 12\), the statement is false.

Key Concepts

Algebraic Inequalities

Understanding algebraic inequalities is crucial for solving various mathematical problems. An inequality, unlike an equation, indicates a relationship where one value is greater than or less than another. It’s no different than comparing two numbers, except one or both of these numbers involves a variable, like 'x'.

Let's consider the given exercise, which involves the inequality \( -\frac{1}{2} x + 6 > 0 \). The inequalities, much like equations, can often be manipulated to isolate the variable. However, the rules of manipulation differ slightly due to the direction of the inequality sign, which adds an additional layer of complexity. While preparing an inequality like this for a solution, it’s helpful to keep in mind that your goal is to find the range of values for ‘x’ that make the inequality true. You do this by performing operations to isolate ‘x’ on one side, which often involves operations such as addition, subtraction, multiplication, or division.

Inequality Manipulation

Manipulating inequalities is a delicate process and must be done with great care to preserve the inequality's integrity. Let's delve into the manipulation done in the provided exercise for greater clarity.

Initially, you want to isolate the variable on one side, resulting in the inequality \( -\frac{1}{2} x > -6 \). Here’s where many students trip up—you’re rightfully tempted to remove the negative coefficient by multiplying or dividing by a negative number. But here’s the kicker: whenever you do that to an inequality, you must flip the inequality sign. This is a fundamental rule in inequality manipulation which is critical for arriving at the correct solution.

By multiplying by -2 to both sides in the exercise, we correctly flip the inequality sign to get \( x < 12 \). This step is vital and can’t be stressed enough; a mistake here would lead to the wrong conclusion about which numbers are part of the solution set. Remember, inequality manipulation involves careful tracking of each step to ensure the final inequality accurately represents the original statement.

Inequality Solution Justification

Rationalizing and justifying the solution to an inequality is as important as finding it. Understanding why a solution is correct or incorrect helps to avoid misconceptions and errors, especially when two inequalities appear to be similar. In the exercise presented, the solution provided need to be justified against the claim that \( -\frac{1}{2} x + 6 > 0 \) is equivalent to \( x > 12\).

Upon manipulation, we reached \( x<12\), which directly contradicts the statement \( x > 12\), as the direction of the inequality does not match. In our justification, we must highlight that the symbol ‘>’ indicates values greater than, whereas the symbol ‘<‘ indicates values less than. They cannot be equivalent as they are fundamentally different statements about the value of ‘x’. It's key to always review and justify your solution, ensuring that each manipulation step was valid and logically supports the final answer.