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Chapter 1: Problem 9

Solve. \(3-\frac{1}{4} x=\frac{3}{2}\)

Short Answer

Expert verified
x=6

Step by step solution

01

Isolate the Variable Term

Subtract 3 from both sides of the equation to get the term with the variable alone on one side.\[3 - \frac{1}{4}x - 3 = \frac{3}{2} - 3\]This simplifies to:\[-\frac{1}{4}x = \frac{3}{2} - 3\]
02

Simplify the Right Side

Convert 3 to a fraction with the same denominator as \(\frac{3}{2}\). Recall that 3 is \(\frac{6}{2}\). So the equation becomes:\[-\frac{1}{4}x = \frac{3}{2} - \frac{6}{2}\]Simplify the right side:\[-\frac{1}{4} x = -\frac{3}{2}\]
03

Solve for x

To isolate x, multiply both sides of the equation by the reciprocal of \(-\frac{1}{4}\), which is \(-4\).\[-4 \times -\frac{1}{4}x = -4 \times -\frac{3}{2}\]Simplify:\[x = 6\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating Variables
When solving algebraic equations, isolating the variable is a crucial step. This means moving all terms containing the variable to one side of the equation and other terms to the opposite side.
This process often involves several actions:
  • Subtracting or adding terms on both sides
  • Combining like terms
Consider the exercise:
3 - \(\frac{1}{4}\) x = \(\frac{3}{2}\).
We isolated the variable term \(-\frac{1}{4}x\) by subtracting 3 from both sides.
The intermediate step looks like this: \[3 - \frac{1}{4}x - 3 = \frac{3}{2} - 3\] This simplifies to:
\[-\frac{1}{4}x = \frac{3}{2} - 3\] We successfully isolated the term containing x.

This process helps make the equation easier to solve.
Fraction Operations
Working with fractions can seem challenging but becomes straightforward with practice. In this exercise, we had to deal with operations involving fractions. Remember these key points:
  • When subtracting fractions, they must have a common denominator
  • Convert whole numbers to fractions to combine them
Let's simplify \(\frac{3}{2} -3\).
Convert 3 to a fraction: \(3 = \frac{6}{2}\).
Now, our equation looks like this:
\[-\frac{1}{4}x = \frac{3}{2} - \frac{6}{2}\].
With the same denominators, we can subtract directly: \[-\frac{1}{4}x = -\frac{3}{2}\].
Understanding these steps aids in working through fraction operations smoothly and accurately.
Reciprocal
The concept of the reciprocal is vital for solving many equations. The reciprocal of a number is 1 divided by that number. For example, the reciprocal of 4 is \(\frac{1}{4}\).
Similarly, the reciprocal of \(-\frac{1}{4}\) is -4.
In our exercise, to isolate x, we multiplied both sides by the reciprocal of \(-\frac{1}{4}\):
\[-4 \times -\frac{1}{4}x = -4 \times -\frac{3}{2}\].
Resulting in:
\[x = 6\].
Recognizing and using reciprocals correctly helps simplify and solve equations involving fractions efficiently.

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