Problem 16 - Algebra Form and Function

Question

Each of the inequalities can be solved by performing two operations on both sides. State the operations in order of use and solve the inequality. $$ 3.7-v \leq 5.3 $$

Step-by-Step Solution

Answer

Answer: The range of values 'v' can take is any value greater than or equal to \(-1.6\).

Step 1: Identifying the Operations

First, we need to identify the operations to perform on both sides of the inequality. From the given inequality \(3.7 - v \leq 5.3\), we need to carry out the following operations in order: 1. Add 'v' to both sides. 2. Subtract '3.7' from both sides.

Step 2: Solving for 'v'

Now, let's solve the inequality by applying the identified operations: 1. Add 'v' to both sides: $$ (3.7 - v) + v \leq 5.3 + v $$ $$ 3.7 \leq 5.3 + v $$ 2. Subtract '3.7' from both sides: $$ 3.7 - 3.7 \leq 5.3 + v - 3.7 $$ $$ 0 \leq 1.6 + v $$

Step 3: Final Answer

We have now solved the inequality and found that: $$ v \geq -1.6 $$ As a result, any value of 'v' greater than or equal to \(-1.6\) will satisfy the given inequality.

Key Concepts

Solving Inequalities

When dealing with inequalities, there's a close similarity to solving equations, but with a crucial difference. Inequalities deal with expressions that express ranges rather than a single value. The inequality sign can take forms such as \(<, >, \leq, \, \text{or} \, \geq \). The main goal in solving inequalities is to find the range of values that satisfy the inequality condition.

For example, consider the inequality \(3.7 - v \leq 5.3\) given above. Our objective here is to isolate the variable \(v\) on one side. This will allow us to understand which values \(v\) can take to make the inequality true.

In real-world terms, solving these inequalities can help us understand limits, constraints, or even safety measures in various scenarios. By determining the range of acceptable solutions, you essentially open a gateway to myriad solutions that satisfy the inequality condition.

Mathematical Operations

Mathematical operations include a variety of functions that can be applied to numbers or expressions. In the context of inequalities, the basic operations include addition, subtraction, multiplication, and division. The operations are performed with the purpose of isolating the variable of interest.

Let's apply this to our inequality \(3.7 - v \leq 5.3\). To get \(v\) by itself, the operations required are as follows:

• Initially, add 'v' to both sides to eliminate \(-v\) from the left.
• Subsequently, subtract '3.7' from both sides to deal with the constant term.

The principle behind these operations is to maintain the balance of the inequality. Adding or subtracting the same value on both sides ensures the inequality holds true throughout the simplification process.

However, one must take care when multiplying or dividing by negative numbers, as this will reverse the inequality sign. In our example, such operations were not needed, but it's essential to remember this rule for other cases.

Algebraic Solution Steps

The process of solving inequalities often involves a sequence of algebraic steps. It is crucial to approach these with a logical framework to ensure clarity and accuracy. The steps taken in solving the inequality \(3.7 - v \leq 5.3\) demonstrate the clean application of algebraic techniques:

• **Step 1**: Add \('v'\) to both sides. This helps in removing the \(-v\) from the left-hand side, simplifying the expression to \(3.7 \leq 5.3 + v\).


• **Step 2**: Subtract \('3.7'\) from both sides. Simplifying the right-hand side gives us \(0 \leq 1.6 + v\).


• **Step 3**: Rearrange the inequality to solve for \(v\). Consequently, we derive \(v \geq -1.6\).

These basic steps highlight the importance of applying consistent operations on both sides of the inequality. When done correctly, this method will yield solutions that validate the original inequality condition.

Understanding and practicing these steps can significantly enhance your ability to solve more complex inequalities in algebra.