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Chapter 3: Problem 67

Solve. $$\sqrt{3 x-5}+\sqrt{2 x+3}+1=0$$

Short Answer

Expert verified
There is no solution.

Step by step solution

01

Identify the problem

Notice that the equation involves square roots and an addition that equals zero: \[ \sqrt{3x-5} + \sqrt{2x+3} + 1 = 0 \]
02

Isolate one square root term

Subtract 1 from both sides of the equation to isolate the square root terms:\[ \sqrt{3x-5} + \sqrt{2x+3} = -1 \]
03

Consider the properties of square roots

The sum of two square root terms should be greater than or equal to zero, but -1 is negative. This leads to a contradiction.
04

Conclude the solution

Since the sum of the square root terms cannot equal a negative number, there is no solution to this equation.

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Key Concepts

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

square root equations
Square root equations are mathematical expressions where the variable is under a square root symbol. For example, \( \sqrt{3x - 5} \) is considered a square root term.
Coping with square root equations means you often need to isolate these terms to solve or simplify the equation.
This usually involves steps like subtracting, adding, or squaring both sides of the equation.

However, it鈥檚 crucial to remember that the principal square root of an expression is always non-negative. This is because the square root function \( \sqrt{y} \) represents only the positive root, not both the positive and negative roots.
This will become important in later steps of the problem.
no solution properties
In mathematics, some equations do not have a solution due to the inherent properties of the expressions involved. Let's explore why this might happen in the context of our problem.

When we examine \( \sqrt{3 x-5} + \sqrt{2 x+3} = -1 \), it is clear from properties of the square root function that both terms \( \sqrt{3x-5} \) and \( \sqrt{2x+3} \) must always be zero or positive.
However, the equation requires their sum to equal -1, which is impossible since the sum of two non-negative numbers cannot produce a negative result.
This is a typical example where an equation has no solution due to conflicting requirements from their basic algebraic properties.
equation contradiction
In algebra, an equation contradiction occurs when the requirements of the equation cannot simultaneously be satisfied. Essentially, it leads to a logical inconsistency within the problem.

In our given problem, we reached the equation \( \sqrt{3x-5} + \sqrt{2x+3} = -1 \).
This tells us that for any real number \( x \), the left side can never be negative because square roots themselves cannot be negative.
Thus, finding the sum of non-negative values to be equal to -1 is a clear contradiction.
Therefore, we conclude that no real number \( x \) satisfies this equation, indicating that the given equation has no solution.

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