Problem 9 - Intermediate Algebra

Question

What is the first step in solving each equation? a. \(\sqrt{x+11}=5\) b. \(\sqrt[3]{5 x+4}+3=30\) c. \(\sqrt{x+8}-\sqrt{2 x}=1\)

Step-by-Step Solution

Answer

The first steps are isolating each root: a - ready, b - subtract 3, c - move a root to the RHS.

Step 1: Isolate the Square Root for Equation a

For the equation \( \sqrt{x+11}=5 \), observe that the square root is already isolated on the left side of the equation. Thus, your first step to solving this equation is complete.

Step 2: Subtract 3 from Both Sides for Equation b

For the equation \( \sqrt[3]{5x+4}+3=30 \), you need to isolate the cube root. Start by subtracting 3 from both sides of the equation to eliminate the 3: \( \sqrt[3]{5x+4} = 27 \).

Step 3: Move One Square Root to the Other Side for Equation c

For the equation \( \sqrt{x+8}-\sqrt{2x}=1 \), rearrange the terms to move one square root to the other side. Add \( \sqrt{2x} \) to both sides: \( \sqrt{x+8} = \sqrt{2x} + 1 \).

Key Concepts

Isolation of Square Roots

When solving an equation with square roots, one of the key steps is to isolate the square root expression. This means getting the square root by itself on one side of the equation so that we can eliminate it to solve for the variable.

• Start by examining the equation. If there are any numbers or terms added to or subtracted from the square root, move them to the other side by adding or subtracting them.
• If necessary, divide both sides of the equation by any coefficients in front of the square root.
• Once isolated, square both sides of the equation to eliminate the square root sign.

For the equation \(\sqrt{x+11}=5\), the square root is already isolated on the left side, so the next step is to square both sides to remove the square root: \((\sqrt{x+11})^2 = 5^2\)
Resulting in: \(x+11=25\).
This simple equation can now be solved easily to find \(x\).

Isolation of Cube Roots

Cube roots in equations need to be isolated just like square roots. Isolating a cube root involves setting it alone on one side so it can be eliminated through algebraic manipulation.

• Look for additional terms added or subtracted from the cube root, and move them to the other side by performing the inverse operation.
• Ensure there are no coefficients attached to the cube root. If there are, divide them out.
• Once isolated, cube both sides of the equation to remove the cube root.

For the equation \(\sqrt[3]{5x+4} + 3 = 30\), begin isolating the cube root by subtracting 3 from both sides: \(\sqrt[3]{5x+4} = 27\).
Now, raise both sides to the power of three: \((\sqrt[3]{5x+4})^3 = 27^3\)
Removing the cube root results in: \(5x+4 = 19683\).
Now, solve for \(x\) with basic algebra.

Algebraic Manipulation

Algebraic manipulation involves rearranging the terms of an equation to make it easier to solve. This is especially crucial when dealing with equations containing multiple radicals.

• Identify the terms that need to remain on one side of the equation and those that need to be shifted to the other side.
• Use basic algebraic operations such as addition, subtraction, multiplication, or division to rearrange the terms.
• Use inverse operations to cancel out terms when necessary.

Consider the equation \(\sqrt{x+8} - \sqrt{2x} = 1\). The first step is to move one of the radicals to the opposite side of the equation. By adding \(\sqrt{2x}\) to both sides, you get: \(\sqrt{x+8} = \sqrt{2x} + 1\).
Now, each side of the equation can be squared to eliminate the square roots, allowing further simplification and eventually, a solution for \(x\).
Remember, precise algebraic manipulation is essential for simplifying complex radical equations to a solvable form.