Problem 1 - Calculus and its applications

Question

Complete each of the following statements. As x approaches______________ , the value of -3x approaches 6.

Step-by-Step Solution

Answer

As \(x\) approaches -2.

Step 1: Understand the expression

The expression given to us is \(-3x\) and we need to find out what value \(x\) should approach so that this expression equals 6. Essentially, we are asked to solve the equation \(-3x = 6\).

Step 2: Rewrite the equation

The equation we have is \(-3x = 6\). We need to solve for \(x\), so our target is to express \(x\) on one side of the equation.

Step 3: Isolate x

To isolate \(x\), we need to divide both sides of the equation by -3. This gives us:\[x = \frac{6}{-3}\]

Step 4: Solve the equation

Carry out the division on the right-hand side:\[x = -2\]This tells us that as \(x\) approaches -2, the expression \(-3x\) approaches 6.

Key Concepts

Solving Equations

Solving equations is a fundamental skill in mathematics, especially in calculus. It is the process of finding the values that satisfy an equation. In the exercise provided, we need to figure out what value of \(x\) makes the equation \(-3x = 6\) true.
To solve such equations, follow these general steps:

• Identify the operation being applied to the variable \(x\) (in our case, multiplication by -3).
• Apply the inverse operation to both sides of the equation to isolate \(x\). In this instance, divide both sides by -3.
• After performing the inverse operations, you find that \(x = -2\).

By approaching the problem systematically, solving becomes straightforward, and it's easier to understand what the equation describes mathematically. When \(x\) is -2, it satisfies the equation so that \(-3x = 6\) is true.

Algebraic Manipulations

Algebraic manipulation involves rearranging expressions or equations to isolate and solve for the variable of interest. These manipulations are vital in calculus where solving limits often relies on such techniques.
In the exercise, manipulating the equation \(-3x = 6\) involves simple steps:

• Recognize the need to isolate \(x\) by doing the opposite of multiplication (i.e., division).
• Perform the division by -3 on both sides which directly gives \(x = -2\).

Remember to always perform the same operation on both sides of the equation to maintain equality. This ensures that the resulting value is correct and truly a solution. Mastering algebraic manipulations simplifies the process and is crucial for tackling more complex problems.

Understanding Expressions

Expressions in mathematics represent values or quantities with numbers, variables, and operations. Understanding expressions is key in solving limits as it helps in evaluating and simplifying problems.
For our exercise, the expression \(-3x\) needs to be evaluated as \(x\) approaches -2. To comprehend this:

• Recognize how \(-3x\) changes as \(x\) varies; here, multiply \(-3\) by the approaching value of \(x\).
• Appreciate that as \(x\) approaches -2, substituting this into the expression \(-3x\) results in 6, matching our desired outcome.

Grasping how expressions work allows you to predict their behavior as variables change. This understanding is a cornerstone for calculus, especially when dealing with limits and approaching values.