Problem 61 - College Algebra

Question

For the following exercises, use a graphing calculator \(x\) . Round the answers to the nearest hundredth. $$ (0.25-0.75)^{2} x-7.2=9.9 $$

Step-by-Step Solution

Answer

The solution is \(x = 68.4\).

Step 1: Simplify the Expression

First, simplify the expression \((0.25 - 0.75)^{2}\).Calculate \(0.25 - 0.75 = -0.5\).Then, square the result: \((-0.5)^2 = 0.25\). Now, the expression becomes:\(0.25x - 7.2 = 9.9\).

Step 2: Isolate the Variable

To find the value of \(x\), first add 7.2 to both sides to isolate terms with \(x\) on one side:\(0.25x - 7.2 + 7.2 = 9.9 + 7.2\).This simplifies to:\(0.25x = 17.1\).

Step 3: Solve for x

Now, solve for \(x\) by dividing both sides by 0.25:\(x = \frac{17.1}{0.25}\).Calculate \(x\): \(x = 68.4\).

Step 4: Final Answer

The final rounded solution to the nearest hundredth is verified as \(x = 68.4\) using a graphing calculator.

Key Concepts

Using a Graphing Calculator

When dealing with mathematical equations, especially linear ones, a graphing calculator can be an invaluable tool. It helps visualize equations and solve them quickly by providing step-by-step solutions with ease.
These devices can graph functions, find intersections, and verify solutions thanks to their powerful computational ability. In our exercise, using a graphing calculator allows you to quickly input the simplified expression and solve it efficiently.
This ensures accuracy and helps in verifying manual calculations, particularly when solving for the variable or displaying the graph of the equation. Incorporating a graphing calculator in your study routine saves time and boosts understanding, making math less daunting.

Algebraic Simplification

Algebraic simplification involves reducing an equation to its simplest terms. This makes it easier to solve, as fewer steps are required to find the solution. Practicing this skill is essential in algebra.
In our exercise, simplifying the expression (0.25 - 0.75) first, resulted in (-0.5). When squared, this becomes 0.25, altering the equation to 0.25x - 7.2 = 9.9.
Simplifying such calculations minimizes potential errors and clarifies the path to the solution. Keep practicing simplification to become faster and more accurate in solving equations. It not only aids problem-solving but also builds a strong foundation in algebra.

Rounding to the Nearest Hundredth

Rounding is an important skill, particularly when dealing with decimal numbers. After you find a solution through algebraic calculations or using a graphing calculator, rounding to the nearest hundredth provides clarity and precision.
To round to the nearest hundredth, look at the digit in the thousandth place. If it is 5 or greater, round up. If it's less than 5, round down. For example, our solution for the exercise was initially calculated as 68.4. Since no extra decimals exist that may alter the value, it remains unchanged when rounded.
This step ensures results are tidy and suitable for further use or presentation. Consistently applying rounding rules will make your results understandable and neat, especially in scientific and financial calculations.