Problem 61 - College Algebra with Corequisite Support

Question

For the following exercises, use a graphing calculator to solve for \(x\). 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 \(0.25 - 0.75\), which gives \-0.5\. Thus, \((0.25 - 0.75)^{2} = (-0.5)^{2} = 0.25\). This simplifies the equation to \(0.25x - 7.2 = 9.9\).

Step 2: Isolate the x-term

Add 7.2 to both sides of the equation to isolate the term containing \(x\): \(0.25x - 7.2 + 7.2 = 9.9 + 7.2\), which simplifies to \(0.25x = 17.1\).

Step 3: Solve for x

Divide both sides of the equation by 0.25 to solve for \(x\): \(x = \frac{17.1}{0.25}\).

Step 4: Calculate the Value of x

Calculate the division to find \(x\): \(x = 68.4\).

Step 5: Round the Answer

Since the problem asks to round to the nearest hundredth, \(x = 68.4\) is already rounded to the nearest hundredth.

Key Concepts

Solving Linear Equations

Linear equations form the backbone of algebra and are important for solving real-world problems. A linear equation is any equation that can be expressed in the form \( ax + b = c \). Here, \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable we need to solve for. Solving this type of equation involves isolating \( x \) to one side by performing algebraic operations that are equivalent on both sides.

In solving linear equations, the key steps are to simplify the equation, combine like terms, and then use inverse operations — like addition or subtraction, and multiplication or division — to isolate the variable. For example, in the solution provided above, we first simplified \( (0.25 - 0.75)^2 \), which gave us a simpler equation: \( 0.25x - 7.2 = 9.9 \). Then, we added 7.2 to both sides, making the equation \( 0.25x = 17.1 \). Finally, dividing both sides by 0.25 gave us \( x = 68.4 \).

Remember, the primary goal in solving a linear equation is to isolate the variable \( x \) using legitimate algebraic techniques. This requires careful manipulation of the equation until it is expressed as \( x = \) some number.

Rounding to Nearest Hundredth

Rounding numbers is an important skill, especially when dealing with decimals. It ensures precision without overcomplicating the answer. Rounding to the nearest hundredth means you want to keep two decimal places. The rule for rounding is simple: look at the third decimal place and decide if you should round up or leave it.

Here’s how it works:

• If the third decimal place is 5 or more, you increase the second decimal place by 1.
• If it's less than 5, you keep the second decimal place as is.

For instance, if we have a number like 68.456, the number at the third decimal place — that’s 5 — suggests rounding the second decimal place up. Thus, this rounds to 68.46. In our problem, however, the calculation gives us 68.4, which already has one decimal place. Since the problem specifically asks for two decimal places, and the original step shows rounding isn’t necessary as 68.40 is technically the same as 68.4 in context, it was already sufficiently rounded.

Step-by-Step Solution

Breaking down a problem into a step-by-step solution helps clarify the process and makes it easier to tackle complex algebra problems.

Let's revisit the process:

• Simplify the equation: Combine and simplify the terms to make the equation more manageable.
• Isolate the variable: Use mathematical operations to get the variable on one side of the equation.
• Solve for the variable: Perform any necessary calculations to find the value of the variable.
• Check your work: Verify that your solution is correct by substituting the variable back into the original equation.
• Final adjustments: Round your answer if necessary, as per the question's requirements.

This approach not only provides a clear path to the solution but also enhances understanding by demonstrating how each step logically follows from the preceding one.