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Chapter 1: Problem 54

Does the equation have a solution? Ex plain how you know without solving it. $$ \frac{x+3}{2 x+5}=1 $$

Short Answer

Expert verified
Answer: Yes, the equation does have a solution, because any value of x other than -5/2 results in a valid and defined equation.

Step by step solution

01

Consider the denominator

The equation will not have a solution if the denominator (2x + 5) results in zero because division by zero is undefined. So, we first need to figure out if there is any value of x that makes the denominator zero. $$ 2x + 5 = 0 $$ Solve for x: $$ 2x = -5 \\ x = -\frac{5}{2} $$
02

Analyze the equation with the critical value

Since we found that there is one value of x (-5/2) which makes the denominator equal to zero, let's see if this value results in an undefined equation or if it is actually a solution. Plugging in x=-5/2 into the equation: $$ \frac{-\frac{5}{2}+3}{2\cdot\left(-\frac{5}{2}\right)+5} = 1 $$
03

Conclusion

Given x = -5/2, the denominator becomes zero: $$ 2\cdot\left(-\frac{5}{2}\right) + 5 = 0 $$ So, the equation is undefined for x = -5/2. However, this does not mean that there is no solution to the equation. The answer to the question "Does the equation have a solution?" is a definitive yes because any value of x other than -5/2 would make the equation defined and could potentially result in a true statement. In other words, with any value of x other than -5/2, we have a valid equation. This means there could indeed be a value of x that makes the equation true, but we are not required to find one for this exercise. Thus, we can conclude that the equation does have a solution without actually solving for the value of x.

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Key Concepts

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

Understanding Denominators in Equations
When dealing with equations like \(\frac{x+3}{2x+5}=1\), it's important to consider the denominator, which is the expression below the fraction line. In this case, the denominator is \(2x + 5\). Denominators play a crucial role in determining whether the fraction, and in turn, the equation, is defined or undefined.
  • If the denominator equals zero for a certain value of \(x\), the fraction becomes undefined, since division by zero is mathematically undefined.
  • To identify these critical values, set the denominator equal to zero and solve for \(x\). For our equation, this looks like: \(2x + 5 = 0\), leading to \(x = -\frac{5}{2}\).
By solving this, we find that the fraction turns undefined when \(x = -\frac{5}{2}\). Remember, for the equation to remain valid, the denominator should not be zero. As a result, when \(x = -\frac{5}{2}\), the entire equation becomes undefined.
Recognizing Undefined Expressions
Undefined expressions occur in equations when a mathematical operation cannot be performed, usually because of division by zero. In the equation \(\frac{x+3}{2x+5}=1\), if \(2x + 5\) equals zero, the expression becomes undefined. Why does this matter?
  • If we substitute \(x = -\frac{5}{2}\) into the equation, the denominator \(2\left(-\frac{5}{2}\right) + 5\) simplifies to zero.
  • This zero denominator indicates that for \(x = -\frac{5}{2}\), the fraction no longer holds meaning in a mathematical context.
Even though \(x = -\frac{5}{2}\) causes an undefined expression, this alone doesn't mean the equation lacks solutions. It simply means the value must be excluded from potential solutions since it leads to a mathematically invalid scenario. Thus, excluding \(x = -\frac{5}{2}\) still allows other values to satisfy the equation.
Solving Algebraic Equations
In algebra, finding solutions to equations involves determining which values of the variable make the equation true. For our equation \(\frac{x+3}{2x+5}=1\), we establish that \(x = -\frac{5}{2}\) cannot be a solution because it makes the equation undefined. Although the problem does not ask us to find the solution explicitly, knowing how to proceed is helpful.
  • Start by rewriting the equation to eliminate the fraction: multiply both sides by the denominator \(2x + 5\).
  • This results in: \(x + 3 = 2x + 5\).
  • Next, solve for \(x\) by simplifying and rearranging terms: \(x + 3 = 2x + 5\Rightarrow 3 = x + 5\Rightarrow x = -2\).
Therefore, while \(x = -\frac{5}{2}\) creates an undefined situation, other values like \(x = -2\) offer valid solutions. Understanding which operations lead to valid mathematical expressions allows us to navigate and solve algebraic equations effectively.

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Most popular questions from this chapter

Let \(f(N)=3 N\) give the number of glasses a cafe should have if it has an average of \(N\) clients per hour. (a) How many glasses should the cafe have if it expects an average of 50 clients per hour? (b) What is the relationship between the number of glasses and \(N ?\) What is the practical meaning of the 3 in the expression \(f(N)=3 N ?\)

In Exercises \(32-37,\) is \(y\) directly proportional to \(x ?\) If so, give the constant of proportionality. $$ y-2=3 x $$

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