Problem 46 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve equation and check your proposed solution. Begin your work by rewriting each equation without fractions. \(\frac{x-2}{3}-4=\frac{x+1}{4}\)

Step-by-Step Solution

Answer

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

Step 1: Remove Fractions

To remove the fractions, multiply every term by the least common multiple (LCM) of 3 and 4, which is 12. So the equation becomes: \[12*(\frac{x-2}{3})-12*4=12*(\frac{x+1}{4})\] Simplifying gives: \[4*(x-2)-48=3*(x+1)\]

Step 2: Simplify the Equation

Simplify each side of the equation: \[4x-8-48=3x+3\] This reduces to: \[4x-56=3x+3\]

Step 3: Isolate x

Get x on one side and numbers on the other. Subtract 3x from both sides and add 56 to both sides: \[4x-3x=3+56\] This simplifies to: \[x=59\]

Step 4: Check the Solution

Substitute 59 for x in the original equation and verify that the equation holds: \[\frac{59-2}{3}-4=\frac{59+1}{4}\] Simplify each side: \[\frac{57}{3}-4=\frac{60}{4}\] Which is equivalent to: \[19-4=15\] So the solution x=59 checks out.

Key Concepts

Fraction Removal

When solving equations that include fractions, a helpful first step is to eliminate them. This process is known as "Fraction Removal." Fractions can often complicate equations and make them more challenging to work with.
To effectively remove fractions, we multiply each term on both sides of the equation by the least common multiple (LCM) of the denominators.

• Identify the denominators in the equation. In our example, these are 3 and 4.
• Determine the LCM of these denominators. The LCM of 3 and 4 is 12.
• Multiply every term in the equation by this LCM.

After doing this, your equation no longer contains fractions, and you can proceed to solve it in standard algebraic form. This step simplifies the equation and reduces the potential for error in subsequent steps.

Equation Simplification

Once the fractions are removed, the next step is to simplify the equation. Simplification means making the equation as straightforward as possible.
In our example, after the fractions were removed, the equation looked like this:

• Begin by distributing any factors across terms. For example, apply the multiplication to terms within parentheses: \(4(x-2)\) becomes \(4x - 8\).
• Combine like terms on the same side of the equation. For example, combine -8 and -48 to get -56.

This method turns the equation into a simpler form of \(4x - 56 = 3x + 3\). When simplified, equations are easier to manipulate, and crucial elements stand out, which aids in solving them efficiently.

Isolating Variable

After simplifying the equation, you should now isolate the variable, typically denoted by \(x\) or another letter. Isolating the variable involves rearranging the equation so that one side only contains the variable you are solving for.
To isolate \(x\):

• First, subtract 3x from both sides to bring all terms containing \(x\) together. This operation gives you \(4x - 3x = x\).
• Next, move the constant terms to the other side by adding or subtracting them. In our example, add 56 to both sides: \(3 + 56 = 59\).

Thus, you find that \(x = 59\). The goal is always to have the variable on one side of the equation, set equal to a number, indicating you're one step closer to finding the solution.

Solution Verification

Verification of the solution is an essential part of algebraic equation solving. This step confirms that your solution is correct and ensures accuracy.
Here's how you can verify a solution:

• Substitute the value of the variable back into the original equation. In our exercise, replace \(x\) with 59.
• Simplify both sides of the equation separately and ensure both sides are equal.

For example, substituting gives you: \(\frac{59-2}{3} - 4\) on one side and \(\frac{59+1}{4}\) on the other.By simplifying, you'll find that both sides equal 15, confirming \(x = 59\) is correct. Verification is crucial as it validates your solution and provides confidence in your result.