Problem 50 Determine the restrictions on \(... [FREE SOLUTION]

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Chapter 1: Problem 50

Determine the restrictions on $x$. $$ \frac{2}{x+1}-\frac{5}{x-7}=\frac{2}{3} $$

Short Answer

Expert verified

The restrictions on $x$ are \[x \eq -1\] and \[x \eq 7\].

Step by step solution

Identify Denominators

First, identify the denominators in the equation. The denominators are $x + 1$ and $x - 7$.

Set Denominators Not Equal to Zero

To determine the restrictions on $x$, set each denominator not equal to zero since division by zero is undefined.

Solve for Each Denominator

1. For $x + 1 \eq 0$: Solve for $x$. \[x + 1 \eq 0\] results in \[x \eq -1\]. 2. For $x - 7 \eq 0$: Solve for $x$. \[x - 7 \eq 0\] results in \[x \eq 7\].

Combine Restrictions

Combine these results to find the complete set of restrictions on $x$: \[x \eq -1\] and \[x \eq 7\].

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Key Concepts

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

Denominators in Fractions

In any fraction, the number under the line鈥攖he denominator鈥攊s crucial. It tells us into how many parts the whole is divided. For instance, in the fraction $ \frac{2}{3} $, the denominator is 3, meaning the whole is divided into three equal parts. When solving equations involving fractions, the denominators play a significant role. They must not be zero, as this would make the fraction undefined.
In the given exercise, the denominators are $ x+1 $ and $ x-7 $. Identifying these denominators is the first step to solving the equation. If any denominator equals zero, the entire fraction becomes undefined.

Division by Zero

One of the fundamental rules in mathematics is that division by zero is undefined. This is because dividing something into zero parts doesn't make sense. When we divide by zero, we face an undefined operation, which mathematical equations cannot handle.
In the equation provided, if $ x+1 $ or $ x-7 $ equals zero, the division will be undefined. To prevent this, you have to set each denominator not equal to zero and solve for $ x $.
For $ x+1 eq 0 $, solving gives $ x eq -1 $.
For $ x-7 eq 0 $, solving gives $ x eq 7 $.
Therefore, the restrictions on $ x $ are that $ x $ cannot be either -1 or 7.

Solving Equations

Solving equations often involves finding the values of variables that make the equation true. In our exercise, solving the equation also involves making sure that the solutions do not violate any mathematical rules, like division by zero.
Let's break it down step by step:

First, identify the denominators: $ x+1 $ and $ x-7 $.

Solving these gives the restrictions $ x eq -1 $ and $ x eq 7 $.
Now we have the complete set of restrictions on $ x $. These restrictions ensure that our solution does not involve division by zero, making the equation valid and solvable.

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91影视

Determine the restrictions on \(x\). $$ \frac{2}{x+1}-\frac{5}{x-7}=\frac{2}{3} $$

Short Answer

Step by step solution

Identify Denominators

Set Denominators Not Equal to Zero

Solve for Each Denominator

Combine Restrictions

Key Concepts

Denominators in Fractions

Division by Zero

Solving Equations

One App. One Place for Learning.

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