Problem 10 - 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. $$ -15.03>s+11.4 $$

Step-by-Step Solution

Answer

Answer: The values of s that satisfy the inequality are any numbers less than -26.43 (s < -26.43).

Step 1: Identify the operation needed to isolate the variable "s"

We are given the inequality: $$ -15.03 > s + 11.4 $$ To isolate the variable "s", we need to subtract 11.4 from both sides of the inequality.

Step 2: Perform the operation and state if the inequality changes direction

We will now subtract 11.4 from both sides of the inequality. Since we are only performing subtraction, the inequality will not change direction.

Step 3: Solve the inequality

We now have the inequality: $$ -15.03 - 11.4 > s $$ On the left side, we subtract the numbers: $$ -26.43 > s $$ So, the solution to the inequality is: $$ s < -26.43 $$ Therefore, the value of the variable "s" that satisfies the inequality is any number less than -26.43.

Key Concepts

Understanding Inequality Direction

When solving inequalities, it's essential to pay close attention to the direction of the inequality symbol. Unlike equations that usually have an equal sign, inequalities use symbols like ">", "<", ">=", or "<=".
These symbols indicate that one side is either greater or less than the other. In the given exercise, the inequality is \(-15.03 > s + 11.4\).Here, the symbol ">" means that the left side is greater than the right side.
Maintaining Direction

• If you add or subtract the same number from both sides of an inequality, the direction remains the same.
• However, multiplying or dividing by a negative number will reverse the direction.

Understanding these rules helps ensure that solution maintains its validity throughout the process.

Isolation of Variable: A Key Step

Isolating the variable is a fundamental step in solving any inequality or equation. The goal is to have the variable on one side of the inequality, ideally by itself.
For the inequality \(-15.03 > s + 11.4\), we want to isolate "s" by getting it alone on one side. This often involves removing any constants or coefficients attached to the variable using basic arithmetic operations.
Steps for Isolation

• Identify the operation connecting the variable with other terms (in this case, addition).
• Perform the opposite operation on both sides (subtraction, here) to eliminate additional terms.
• Always perform the operation across the entire inequality to keep it balanced.

This skill of isolating a variable is important in both mathematics and real-world problem-solving scenarios.

Subtraction Operation in Inequalities

Subtraction is a common operation used to isolate variables in inequalities. It helps in removing constants from one side of the inequality to free up the variable.
In our given problem, we need to subtract 11.4 from both sides to isolate "s". Performing this operation results in the inequality shifting to \(-15.03 - 11.4 > s\), simplifying to \(-26.43 > s\).
Key Points about Subtraction

• Subtraction does not change the direction of the inequality.
• It must be applied equally to both sides to maintain the integrity of the inequality.
• The result needs to be checked so that no errors were made during the subtraction process.

Subtraction is straightforward, but understanding its effect in inequalities is crucial for the correct solution.