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Chapter 1: Problem 28

$$\text { Solve each equation analytically. Check it analytically, and then support your solution graphically.}$$ $$\frac{x-2}{4}+\frac{x+1}{2}=1$$

Short Answer

Expert verified
Solution: \(x = \frac{4}{3}\); verified analytically and graphically.

Step by step solution

01

Clear the Fractions

To eliminate the fractions, multiply every term by 4, which is the least common multiple of 4 and 2. This gives us: \[ 4 \times \frac{x-2}{4} + 4 \times \frac{x+1}{2} = 4 \times 1 \] This simplifies to: \[ x-2 + 2(x+1) = 4 \]
02

Simplify and Combine Like Terms

Distribute the 2 in the equation: \[ x - 2 + 2x + 2 = 4 \] Now, combine the like terms: \[ 3x = 4 \]
03

Solve for x

To isolate \(x\), divide both sides by 3: \[ x = \frac{4}{3} \]
04

Check the Solution Analytically

Substitute \(x = \frac{4}{3}\) back into the original equation to verify: The original equation is: \[ \frac{x-2}{4} + \frac{x+1}{2} = 1 \] Substitute \(x = \frac{4}{3}\) into this equation: \[ \frac{\frac{4}{3}-2}{4} + \frac{\frac{4}{3}+1}{2} \] Simplifying each term gives: \[ \frac{\frac{-2}{3}}{4} + \frac{\frac{7}{3}}{2} \rightarrow \frac{-2}{12} + \frac{7}{6} \rightarrow -\frac{1}{6} + \frac{7}{6} = 1 \] The solution checks out.
05

Support the Solution Graphically

To graphically support the solution, plot the left-hand side and right-hand side of the equation as functions of \(x\). The left side, \( \frac{x-2}{4} + \frac{x+1}{2} \), intersects the constant function \(y=1\) at \(x = \frac{4}{3}\). This confirms the analytical solution.

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Key Concepts

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

Solving Equations
Solving equations is a foundational skill in college algebra. It involves finding the value of a variable that makes the equation true. For example, given the equation \(\frac{x-2}{4}+\frac{x+1}{2}=1\), the goal is to determine which value of \(x\) satisfies it. In this exercise, we start by removing fractions. This makes the equation easier to handle by transforming it into a simpler form.
Here are some simple steps to follow when solving equations:
  • Identify and eliminate any fractions by multiplying every term by the least common multiple of the denominators.
  • Simplify the equation by combining like terms. This involves grouping all similar variable terms and constant terms together.
  • Isolate the variable on one side of the equation to find its value.
Once simplified, check the solution by substituting it back into the original equation to ensure that both sides are equal. This verification ensures accuracy in your results.
Fractions in Equations
Dealing with fractions in equations can seem complicated, but it's manageable with a structured approach. In equations like \(\frac{x-2}{4}+\frac{x+1}{2}=1\), handling fractions is often the first step.
To eliminate fractions:
  • Find the least common multiple (LCM) of the denominators. In this case, the LCM of 4 and 2 is 4.
  • Multiply every term in the equation by this LCM. This step clears the fractions, leaving you with a simpler, whole-number equation.
  • With the fractions gone, proceed to simplify and solve the equation as you would with any other linear equation.
This process of clearing fractions not only simplifies the solving process but also reduces the potential for errors when performing algebraic manipulations. Thorough practice with these techniques will improve your confidence in handling equations with fractions.
Graphical Solutions
Graphical solutions offer a visual way to understand and verify algebraic equations. After solving an equation analytically, like \(\frac{x-2}{4}+\frac{x+1}{2}=1\), we can use graphs to confirm our findings.
Here's how you can check your solution graphically:
  • Graph both sides of the equation as separate functions. For our equation, you graph \(y=\frac{x-2}{4} + \frac{x+1}{2}\) and \(y=1\).
  • Look for the point where these graphs intersect. This intersection point indicates the solution.
  • In our example, the intersection occurs at \(x=\frac{4}{3}\), which is the solution we found analytically.
Graphing ensures that the analytical solution is correct and provides a visual demonstration of how equations behave. It can also help identify any errors made during analytical solutions and solidify the connection between numerical and graphical representations of algebra.

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Most popular questions from this chapter

The table shows equivalent temperatures in degrees Celsius and degrees Fahrenheit. $$\begin{array}{c|c|c|c|c|c}\circ \mathrm{F} & -40 & 32 & 59 & 95 & 212 \\\\\hline^{\circ} \mathrm{C} & -40 & 0 & 15 & 35 & 100\end{array}$$ (a) Plot the data by having the \(x\) -axis correspond to Fahrenheit temperature and the \(y\) -axis to Celsius temperature. What type of relation exists between the data? (b) Find a function \(C\) that uses the Fahrenheit temperature \(x\) to calculate the corresponding Celsius temperature. Interpret the slope. (c) What is a temperature of \(83^{\circ} \mathrm{F}\) in degrees Celsius?

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Using the variable \(x\), write each interval using set-builder notation. $$(3, \infty)$$
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