Problem 46 - Precalculus Student Solutions Manual 5th

Question

Solve each equation. $$\sqrt{x+5}+2=\sqrt{x-1}$$

Step-by-Step Solution

Answer

x = 3.25

Step 1: Isolate one of the square root terms

Start by isolating one of the square root terms. Subtract 2 from both sides of the equation to isolate \(\sqrt{x+5}\): \[ \sqrt{x+5} = \sqrt{x-1} - 2 \]

Step 2: Square both sides of the equation

To eliminate the square roots, square both sides of the equation: \[ \left( \sqrt{x+5} \right)^2 = \left( \sqrt{x-1} - 2 \right)^2 \] Simplifying both sides, we get: \[ x + 5 = x - 1 + 4 \sqrt{x-1} + 4 \]

Step 3: Simplify the equation

Subtract x from both sides: \[ 5 = -1 + 4 \sqrt{x-1} + 4 \]\ \ Rearrange and combine like terms: \[ 6 = 4 \sqrt{x-1} \]

Step 4: Isolate the remaining square root

Divide both sides by 4 to isolate the square root: \[ \sqrt{x-1} = \frac{6}{4} = 1.5 \]

Step 5: Square both sides again

Square both sides again to solve for x: \[ \left(\sqrt{x-1}\right)^2 = \left(1.5\right)^2 \] Simplify: \[ x - 1 = 2.25 \]

Step 6: Solve for x

Finally, add 1 to both sides of the equation: \[ x = 3.25 \]

Key Concepts

isolating square roots

When solving square root equations, the first step is to isolate at least one of the square root terms. This makes it easier to work with and eventually eliminate.

In our example, we start with: \( \sqrt{x+5}+2=\sqrt{x-1} \) To isolate the term \sqrt{x+5}\, we need to get rid of the 2 on the left-hand side. Subtract 2 from both sides of the equation: \( \sqrt{x+5} = \sqrt{x-1} - 2 \) Now, the equation is simpler, and one of the square roots stands alone on one side.

squaring both sides

After isolating one of the square roots, the next step is to eliminate the square roots by squaring both sides of the equation. This process helps to remove the square root symbols and transforms the equation into a polynomial form.

In our case, squaring both sides of the equation \( \sqrt{x+5} = \sqrt{x-1} - 2\) results in: \( (\sqrt{x+5})^2 = (\sqrt{x-1} - 2)^2 \) Simplifying both sides, we get: \( x + 5 = (x - 1) + 4 \sqrt{x-1} + 4 \) Now, we have transformed the equation into a form where we can work with like terms.

combining like terms

Combining like terms is an essential step in solving equations. It means grouping similar terms together to simplify the equation.

Starting from the simplified equation: \( x + 5 = x - 1 + 4 \sqrt{x-1} + 4 \) First, subtract x from both sides: \( 5 = -1 + 4 \sqrt{x-1} + 4 \) Then, combine the constants on the right side: \( 5 = 3 + 4 \sqrt{x-1} \) Subtract 3 from both sides to further simplify: \( 2 = 4 \sqrt{x-1} \) Finally, divide by 4 to isolate the remaining square root: \( \sqrt{x-1} = \frac{2}{4} = 0.5 \)

solving for x

Now, we must solve for x by eliminating the remaining square root. This involves a few more steps to get to our solution.

With the isolated square root term \( \sqrt{x-1} = 0.5 \), square both sides again to remove the square root: \( (\sqrt{x-1})^2 = (0.5)^2 \) Simplify both sides to: \( x - 1 = 0.25 \) Finally, add 1 to both sides to solve for x: \( x = 0.25 + 1 = 1.25 \) And there you have it! Our solution for x is \( x = 1.25 \).

Remember, always check your solution by plugging it back into the original equation to ensure it satisfies both sides!