Problem 8 - 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. $$ -5 t<17.5 $$

Step-by-Step Solution

Answer

Question: Solve the inequality -5t > 17.5 and provide the solution. Answer: The solution to the inequality is t > -3.5.

Step 1: Identify the Operation

In this step, we identify the operation that will help us isolate the variable 't'. We will divide both sides of the inequality by -5 to isolate 't'.

Step 2: Perform the Operation & Change the Inequality Direction

As we are dividing by a negative number (-5), the inequality will change direction. Divide both sides of the inequality by -5: $$ \frac{-5t}{-5} > \frac{17.5}{-5} $$

Step 3: Simplify the Inequality

Simplify the inequality and obtain the solution: $$ t > -3.5 $$ The solution to the inequality is \(t > -3.5\).

Key Concepts

Inequality Operations

Solving inequalities is very similar to solving equations. However, one key thing to remember is how operations affect the inequality sign. Whenever you perform an operation on one side, you must perform the same operation on the other side to maintain balance.
Common operations include:

• Adding or subtracting a number from both sides
• Multiplying or dividing both sides by a positive number
• Multiplying or dividing both sides by a negative number

In the example of the inequality \(-5t < 17.5\), the operation needed to solve it requires dividing both sides by -5. Each step in solving an inequality requires careful assessment to determine how it alters the inequality. Proper execution of these operations will maintain mathematical correctness and lead you to the solution.

Changing Inequality Direction

A unique feature of inequalities is that dividing or multiplying both sides by a negative number reverses the inequality sign. This is unlike regular equations, where the order of the terms doesn't matter.
This change of direction ensures that the inequality still represents the same range of values but in the opposite sense. For example, in the inequality \(-5t < 17.5\), dividing both sides by -5 changes the '<' symbol to '>', making it \(t > -3.5\).
It's essential not to overlook changing the direction of the inequality during your calculations, as it directly affects the validity of your solution and can lead to incorrect results if ignored. Always double-check your operations with inequalities to ensure you're applying this rule when necessary.

Isolation of Variables

When dealing with both equations and inequalities, one of the main goals is to isolate the variable. This means you want the variable on one side of the inequality and everything else on the other.
In \(-5t < 17.5\), isolating 't' involves dividing both sides by -5. The result, \(t > -3.5\), shows 't' is isolated successfully. This gives a clear statement about all the possible values that 't' can take.
The isolation process often includes:

• Moving terms to opposite sides of the inequality
• Combining like terms
• Using arithmetic operations to simplify expressions

By mastering the isolation of variables, you'll make solving inequalities more straightforward and efficient, ensuring accuracy and clarity in your resulting solutions.