Problem 17 - Algebra for College Students

Question

Solve each equation. Don't forget to check each of your potential solutions. \(\sqrt{2 x-5}=-1\)

Step-by-Step Solution

Answer

The equation has no real solutions.

Step 1: Understand the Equation

The given equation is \(\sqrt{2x - 5} = -1\). This means that the square root of the expression \(2x - 5\) is equal to \(-1\). By definition, the square root of a number is always non-negative (positive or zero), so it cannot equal a negative number.

Step 2: Analyze the Possibility of a Solution

Since there is no real number whose square root is negative (because the square root function always yields a non-negative result), \(\sqrt{2x - 5}\) must result in a non-negative number. Thus, \(\sqrt{2x - 5} = -1\) has no solution.

Step 3: Conclude the Solution

Given that the square root function cannot produce a negative result, we conclude there are no values of \(x\) that satisfy \(\sqrt{2x - 5} = -1\). Therefore, the equation has no real solutions.

Key Concepts

Square Root Properties

Square roots are very important in math. They play a big role when solving equations. A square root of a number is a value that, when multiplied by itself, gives that number. For example, the square root of 9 is 3 because 3 times 3 equals 9. A key property of square roots is that they are always non-negative. This means they can be zero or a positive number, but never negative.
In symbols, for any real number \(a\), \( \sqrt{a} \geq 0 \).

• If \(a\) is a positive number, \( \sqrt{a} \) will give a positive result.
• If \(a\) equals zero, \( \sqrt{a} \) is just 0.
• However, if \(a\) is negative, \( \sqrt{a} \) is not a real number, but instead is imaginary.

It is this fundamental property which led to the conclusion in the provided equation that there is no real solution.

No Solution Equations

Some equations just do not have solutions. This happens when the conditions set by the equation cannot be met. The equation given, \(\sqrt{2x - 5} = -1\), is a perfect example of this.

Because a square root can never be negative, the equation asks us to find a value for \(x\) that is impossible. This is because there isn't any \(x\) that can make \(2x - 5\) such that its square root is negative.
So, we say the equation has "no solution." It's an important step in math to recognize when an equation isn't solvable, as it helps avoid unnecessary calculations and alerts us to impossible scenarios.

Concept of Real Numbers

Real numbers are a vast collection of numbers used in everyday life. They include natural numbers like 1, 2, and 3, whole numbers like 0, integers like -1, fractions, decimals, and numbers like \( \pi \) and \( \sqrt{2} \).

A critical aspect of real numbers we need to understand for solving equations is that within them, the square roots of negative numbers do not exist. Let's break it down:

• Real numbers are situated on a continuous line without any gaps. This is called the real number line.
• For real number equations, if solving leads to a scenario where a number must be negative under a square root, it's no longer part of the real numbers; it's considered imaginary.

When dealing with equations like \( \sqrt{2x - 5} = -1\), acknowledging the bounds of real numbers clarifies why some equations cannot be solved with real numbers.