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Chapter 6: Problem 58

Solve and check each equation. \(\frac{x-2}{3}-4=\frac{x+1}{4}\)

Short Answer

Expert verified
The solution to the equation \(\frac{x - 2}{3} - 4 = \frac{x + 1}{4}\) is x = 59.

Step by step solution

01

Clear Fractions

First, multiply every term by the least common multiple (LCM) of 3 and 4 to eliminate the fractions. The LCM of 3 and 4 is 12, so the equation becomes \(4(x - 2) - 48 = 3(x + 1)\). After expanding the brackets, we get \(4x - 8 - 48 = 3x + 3\). Simplifying it, we have \(4x - 56 = 3x + 3\).
02

Solve for x

Next, isolate x by subtracting 3x from both sides to have the equation as: \(4x - 3x = 3 + 56\), simplifying it we have \(x = 59\)
03

Check the Answer

Finally, replace x with 59 into the original equation and confirm if both sides are equal. The original equation is \(\frac{x - 2}{3} - 4 = \frac{x + 1}{4}\). Substituting x = 59, we get \(\frac{59 - 2}{3} - 4 = \frac{59 + 1}{4}\) which simplifies to \(19 - 4 = 15\) and hence verifies our solution.

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Key Concepts

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

Understanding Fractions in Equations
Fractions are a way of expressing a part of a whole number. In equations, fractions can sometimes make the problem appear more complex. However, understanding how to work with them effectively can simplify the process.

In many equations, you'll encounter terms like \( \frac{x-2}{3} \) and \( \frac{x+1}{4} \). These represent divisions where the numerator is the expression above the line and the denominator is the number below.
  • To simplify fractions in equations, you can eliminate them by finding a common denominator or by multiplying through by the least common multiple (LCM) of the denominators.
  • This step converts the equation into one without fractions, making it simpler to handle.
By transforming fractions this way, the original problem becomes easier to see and solve.
Finding the Least Common Multiple
To make equations with fractions simpler, it's helpful to eliminate the fractions. This is done using the Least Common Multiple, or LCM, of the denominators.

The LCM is the smallest number that is a multiple of each denominator. To find the LCM of numbers like 3 and 4:
  • List the multiples of each number:
    • Multiples of 3: 3, 6, 9, 12, 15, ...
    • Multiples of 4: 4, 8, 12, 16, 20, ...
  • Identify the smallest common multiple: In this case, it is 12.
Using the LCM allows you to clear fractions from equations by multiplying each term by the LCM. This helps in transforming the equation into a simpler, fraction-free form.
The Importance of Checking Solutions
After solving an equation, it's crucial to verify that the solution is correct by plugging it back into the original equation. This process ensures that no mistakes were made along the way.

Checking solutions involves substituting the found value back into the original equation to see if both sides of the equation balance out.
  • Substitute the value of the variable back into the equation.
  • Simplify both sides to check if they equal each other.
For example, after solving for \( x = 59 \), you substitute it back into the original equation, \( \frac{x-2}{3} - 4 = \frac{x+1}{4} \). If both sides equal, the solution is correct. This step confirms the accuracy of your work and reassures you that the solution is indeed true.

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Most popular questions from this chapter

Explain how to multiply two binomials using the FOIL method. Give an example with your explanation.

Use FOIL to find the products in Exercises 1-8. \((2 x-9)(7 x-4)\)

Solve the equations in Exercises 53-72 using the quadratic formula. \(x^{2}+8 x+12=0\)

I solved \(-2 x+5 \geq 13\) and concluded that \(-4\) is the greatest integer in the solution set.

It's easy to factor \(x^{2}+x+1\) because of the relatively small numbers for the constant term and the coefficient of \(x\).
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