Problem 80 - College Algebra with Modeling and Visualization

Question

Solve the inequality. Approximate the endpoints to the nearest thousandth when appropriate. $$ 5<4 x-2.5 $$

Step-by-Step Solution

Answer

The solution is \((1.875, \infty)\).

Step 1: Isolate the variable term

Add \(2.5\) to both sides of the inequality to eliminate the constant on the right side: \[5 + 2.5 < 4x \]Simplifying the left side will give us:\[7.5 < 4x\]

Step 2: Solve for the variable

To isolate \(x\), divide both sides of the inequality by \(4\):\[\frac{7.5}{4} < x\]This simplifies to approximately:\[1.875 < x\]

Step 3: Express the solution

The inequality solution can be expressed by saying \(x\) is greater than approximately \(1.875\). In interval notation, this can be written as: \((1.875, \infty)\).

Key Concepts

Algebra and Inequality Solving

Solving inequalities involves algebraic techniques to find a range of possible values for a variable. When dealing with algebra, the goal is similar to solving equations, but instead of finding one solution, we're interested in a set of solutions.
For the inequality given, \(5 < 4x - 2.5\), we aim to find when this condition is true that the expression on one side is always less than the expression with the variable.
The key is to manipulate the inequality just like an equation, by doing the same operation on both sides.
Remember:

• When adding or subtracting terms from both sides of an inequality, the inequality symbol remains the same.
• When multiplying or dividing by a positive number, the inequality's direction stays the same.
• If you multiply or divide by a negative number, the inequality's direction will flip.

This process ensures that the inequality remains true while adjusting the variable's position to make it easier to find its possible values.

Isolating the Variable

Isolating the variable is a foundational skill in algebra. It means getting the variable by itself on one side of the equation or inequality. This helps to determine its possible values.
In our exercise, we had initially \(5 < 4x - 2.5\).
To isolate \(x\), first, we wanted to remove the \(-2.5\) from the right side by adding \(2.5\) to both sides. This action balanced the inequality without changing its truth.
Then we have \(7.5 < 4x\).
The next step was to isolate \(x\) by dividing both sides by \(4\), giving us \(\frac{7.5}{4} < x\).
Remember: Always perform operations step by step, and check if you need to reverse the inequality sign if working with negative numbers.

Interval Notation

Interval notation is a concise way of describing a set of numbers along the number line. It's especially useful in expressing solutions to inequalities.
For the solution \(1.875 < x\), this means \(x\) is any number greater than \(1.875\).
In interval notation, this is written as \((1.875, \infty)\).
The round bracket "(" indicates that \(1.875\) is not included in the range. The infinity symbol \(\infty\) always has a parenthesis because it's not a specific number that can be reached.
Always:

• Use round brackets \(()\) for non-inclusive ("open") endpoints.
• Use square brackets \([]\) for inclusive ("closed") endpoints.
• Use comma to separate the lower and upper bounds.

This method delivers clarity and precision, making it easier to see the range of acceptable values for the variable.