Question

a) Solve the equation \(2 \sqrt{3 x+5}+7=16, x \geq-\frac{5}{3}\) algebraically. b) Show how you can use the graph of the function \(y=2 \sqrt{3 x+5}-9, x \geq-\frac{5}{3}\) to find the solution to the equation in part a).

Short answer

a) \( x = \frac{61}{12} \). b) Find where the graph of \( y = 2 \sqrt{3x+5} - 9 \) intersects y = 7.

Step-by-step solution

Isolate the Square Root Term

Start by subtracting 7 from both sides of the equation to isolate the square root term: \[ 2 \sqrt{3x + 5} + 7 = 16 \] becomes \[ 2 \sqrt{3x + 5} = 9 \]

Divide by 2

Next, divide both sides by 2 to further isolate the square root: \[ \sqrt{3x + 5} = \frac{9}{2} \]

Square Both Sides

Square both sides of the equation to eliminate the square root: \[ (\sqrt{3x + 5})^2 = \left( \frac{9}{2} \right)^2 \] This simplifies to \[ 3x + 5 = \frac{81}{4} \]

Solve for x

Isolate x by first subtracting 5 from both sides: \[ 3x + 5 - 5 = \frac{81}{4} - 5 \] This simplifies to \[ 3x = \frac{81}{4} - \frac{20}{4} = \frac{61}{4} \] Finally, divide both sides by 3: \[ x = \frac{61}{4} \div 3 = \frac{61}{12} \]

Verify the Solution

Check that the solution \( x = \frac{61}{12} \) satisfies the original equation and the condition \( x \geq -\frac{5}{3} \).

Using the Graph

Rewrite the given function for part b) to match the form in part a): \[ y = 2 \sqrt{3x+5} - 9 \] Set the function equal to 7 (since \(7 + 9 = 16\)): \[ 2 \sqrt{3x+5} - 9 = 7\] Adding 9 to both sides, we get our original equation: \[ 2 \sqrt{3x+5} = 16 \] By plotting the graph of \(y = 2 \sqrt{3x+5} - 9\) and finding the intersection where y = 7, you will find the same x-value solution that solves the equation.

Key concepts

Algebraic Solution

Solving square root equations algebraically often involves isolating the square root term first. In the given equation, we have:
\[ 2 \sqrt{3x + 5} + 7 = 16 \] Let's isolate the square root term:
Subtract 7 from both sides:
\[ 2 \sqrt{3x + 5} = 9 \] Next, divide by 2:
\[ \sqrt{3x + 5} = \frac{9}{2} \]
To get rid of the square root, you need to square both sides of the equation:
\[ (\sqrt{3x + 5})^2 = \left( \frac{9}{2} \right)^2 \]
This simplifies to:
\[ 3x + 5 = \frac{81}{4} \]
Now, isolate x:
Subtract 5 from both sides:
\[ 3x = \frac{81}{4} - \frac{20}{4} = \frac{61}{4} \]
Lastly, divide by 3:
\[ x = \frac{61}{12} \]
After solving for x, always check your solutions by plugging them back into the original equation. This step is crucial to avoid extraneous solutions that may arise during squaring.

Graphical Solution

Using graphical methods to solve equations provides a visual way to find solutions. From the given equation: \[ 2 \sqrt{3x+5} - 9 = 7 \]
We can rearrange it to align with the original equation:
\[ 2 \sqrt{3x+5} = 16 \]
To solve this graphically, we plot \[ y = 2 \sqrt{3x+5} - 9 \] and look for the intersection where \[ y = 7 \].
Here's how:

• Plot the function \[ y = 2 \sqrt{3x+5} - 9 \].
• Draw a horizontal line at \[ y = 7 \].
• The intersection point of the function and the line gives the x-value that solves the equation.

By mapping the intersection, students can visually confirm the solution they derived algebraically.

Equation Verification

It's essential to verify solutions to avoid mistakes. After finding \[ x = \frac{61}{12} \], substitute this x back into the original equation:
\[ 2 \sqrt{3 \left( \frac{61}{12} \right) + 5} + 7 = 16 \]
Simplify inside the square root:
\[ 2 \sqrt{\frac{183}{12} + 5} + 7 = 16 \]
Convert 5 to a fraction:
\[ 2 \sqrt{\frac{183}{12} + \frac{60}{12}} + 7 = 16 \]
Further simplify:
\[ 2 \sqrt{\frac{243}{12}} + 7 = 16 \]
\[ 2 \sqrt{\frac{81}{4}} + 7 = 16 \]
\[ 2 \left( \frac{9}{2} \right) + 7 = 16 \]
This verifies that \[ x = \frac{61}{12} \] is indeed a valid solution. Checking solutions helps ensure accuracy and reinforces algebraic understanding for students.

Function Transformation

Understanding how functions transform can simplify solving equations. The given function:
\[ y = 2 \sqrt{3x+5} - 9 \]
Includes a square root, a multiplication, and a subtraction.
Here are the key transformations:

• The term inside the square root \[ 3x+5 \] affects the horizontal shift and scaling.
• The coefficient 2 scales the function vertically.
• The subtraction of 9 shifts the function downwards by 9 units.

Recognizing these transformations aids in graphing, simplifying, and solving complex equations. It allows a deeper understanding of how each part affects the graph and ultimately helps find solutions more efficiently.