Problem 37 - Algebra 2 and Trigonometry

Question

In \(3-38,\) solve each equation for the variable, check, and write the solution set. $$ \sqrt{x+4}-\sqrt{x-1}=1 $$

Step-by-Step Solution

Answer

The solution set is \( \left\{ \frac{13}{4} \right\} \).

Step 1: Isolate one square root

Move all terms involving square roots to one side of the equation to isolate one of the roots. The equation is already in such a form: \( \sqrt{x+4} - \sqrt{x-1} = 1 \). We have isolated \( \sqrt{x+4} \).

Step 2: Square both sides

To eliminate the square root, square both sides of the equation:\[(\sqrt{x+4})^2 = (1 + \sqrt{x-1})^2\]This results in:\[ x + 4 = 1 + 2\sqrt{x-1} + (x-1)\]Simplify the right side:\[x + 4 = x + 1 + 2\sqrt{x-1}\]

Step 3: Simplify and isolate the remaining square root

Subtract \(x + 1\) from both sides:\[4 - 1 = 2\sqrt{x-1}\]\[3 = 2\sqrt{x-1}\]Divide both sides by 2 to isolate the square root:\[\frac{3}{2} = \sqrt{x-1}\]

Step 4: Square both sides again

To eliminate the square root, square both sides:\[\left(\frac{3}{2}\right)^2 = (\sqrt{x-1})^2\]This results in:\[\frac{9}{4} = x - 1\]

Step 5: Solve for the variable

Add 1 to both sides to solve for \(x\):\[\frac{9}{4} + 1 = x\]Convert 1 to a fraction with common denominator:\[\frac{9}{4} + \frac{4}{4} = x\]\[\frac{13}{4} = x\]

Step 6: Verify the solution

Substitute \(x = \frac{13}{4}\) back into the original equation to verify:\[\sqrt{\frac{13}{4} + 4} - \sqrt{\frac{13}{4} - 1} = 1\]Calculate each square root:\[\sqrt{\frac{13}{4} + \frac{16}{4}} = \sqrt{\frac{29}{4}} = \frac{\sqrt{29}}{2}\]\[\sqrt{\frac{13}{4} - \frac{4}{4}} = \sqrt{\frac{9}{4}} = \frac{3}{2}\]Confirm the equation holds:\[\frac{\sqrt{29}}{2} - \frac{3}{2} = 1\]Since both sides equal 1, the solution is correct.

Step 7: Write the solution set

The solution set is the value of \(x\) that satisfies the original equation. Therefore, the solution set is:\[\left\{ \frac{13}{4} \right\}\]

Key Concepts

Solution Set

A solution set is a collection of numbers that satisfy an equation. When solving equations with square roots, careful checking is necessary to confirm which number from the set correctly solves the equation.

Consider the equation given in the exercise:

• Our task is to solve for the variable, which in this case is \(x\).
• Usually, after following the proper steps to solve the equation, we arrive at a final value that satisfies the equation when substituted back into the original form. This correct value forms our solution set.

In this specific example, the solution set for the equation \(\sqrt{x+4} - \sqrt{x-1}=1\) is \(\left\{ \frac{13}{4} \right\}\), as it makes the equation true when verified. Knowing the solution set is crucial since it helps confirm that our solution is valid from the range of possible solutions.

Square Both Sides

Squaring both sides of the equation is a powerful technique used to remove square roots. It helps transition an equation from one form to another that is more manageable for further simplifications.

Let's break down the process:

• We start with an isolated square root, like \(\sqrt{x+4}\), which can be squared to remove the square root.
• It's important to note that when we square both sides of an equation, we must consider any additional terms on one or both sides, as this can introduce extra elements in the equation.

For example, when squaring the equation \((\sqrt{x+4})^2 = (1 + \sqrt{x-1})^2\), it evolves to \(x + 4 = 1 + 2\sqrt{x-1} + (x-1)\). Squaring both sides is essential but requires careful handling to avoid errors introduced by new terms.

Isolate the Square Root

Isolating the square root is a strategic step to make the equation ready for squaring. This involves rearranging the equation to have one square root term on one side and everything else on the other.

Here’s how you can isolate the square root:

• Shift all non-square root terms to the opposite side.
• By focusing on one square root, such as \(\sqrt{x+4}\), it allows for a simpler target when squaring.

In the exercise provided, the equation \(\sqrt{x+4} - \sqrt{x-1} = 1\) is already conveniently set up with one square root isolated. This preparation aids in the effective use of squaring both sides, making the solving process more streamlined.

Verify the Solution

Verifying the solution is a critical last step in solving equations with square roots. This step ensures the accuracy and validity of the solution found.

Why verification is crucial:

• Square roots inherently introduce possible extraneous solutions during solving.
• By substituting the solution back into the original equation, we can check if the equation holds true, confirming its correctness.

In the given exercise, substituting \(x = \frac{13}{4}\) back into the original equation\(\sqrt{x+4} - \sqrt{x-1} = 1\) resulted in a valid expression, where both sides equated to 1 when evaluated. Therefore, the verification step confirms that our solution is indeed correct and belongs to the solution set.