Problem 33 - A Graphical Approach to Precalculus with Limits

Question

Use an analytic method to solve each equation in part (a). Support the solution with a graph. Then use the graph to solve the inequalities in parts (b) and (c). (a) \(\sqrt{3 x+7}=2\) (b) \(\sqrt{3 x+7}>2\) (c) \(\sqrt{3 x+7}<2\)

Step-by-Step Solution

Answer

(a) \(x = -1\), (b) \((-1, \infty)\), (c) \(( -\infty, -1)\).

Step 1: Solve the Equality

To solve the equation \( \sqrt{3x + 7} = 2 \), first square both sides to eliminate the square root. This gives us:\[(\sqrt{3x + 7})^2 = 2^2\]This simplifies to:\[3x + 7 = 4\]Next, isolate \(x\) by subtracting 7 from both sides:\[3x = 4 - 7 = -3\]Then, divide by 3 to solve for \(x\):\[x = \frac{-3}{3} = -1\]Thus, the solution to the equation is \(x = -1\).

Step 2: Graph the Function

Plot the function \(f(x) = \sqrt{3x + 7}\) on a graph. The critical point is where \(f(x) = 2\), corresponding to \(x = -1\) which results in the horizontal line \(y = 2\). This graph helps visualize where the function meets the horizontal line at value 2.

Step 3: Analyze the Inequality \( \sqrt{3x + 7} > 2 \)

For the inequality \( \sqrt{3x + 7} > 2 \), we consider the regions where the graph is above the line \(y = 2\). Since the only point where they meet is \(x = -1\), \( \sqrt{3x + 7} > 2 \) holds true for any \(x > -1\). This means the solution to this inequality is the interval \((-1, \infty)\).

Step 4: Analyze the Inequality \( \sqrt{3x + 7} < 2 \)

For the inequality \( \sqrt{3x + 7} < 2 \), look for regions where the graph of \(\sqrt{3x + 7}\) is below the line \(y = 2\). Since \(x = -1\) is the point where the function equals 2, \( \sqrt{3x + 7} < 2 \) holds true for any \(x < -1\). Therefore, the solution to this inequality is the interval \(( -\infty, -1)\).

Key Concepts

Inequalities

Inequalities can seem tricky, but they are just a way of comparing two expressions. Instead of saying two sides are exactly equal, inequalities tell us when one side is less than or greater than the other. There are several types of inequalities:

• Greater than (\( > \))
• Less than (\( < \))
• Greater than or equal to (\( \geq \))
• Less than or equal to (\( \leq \))

In our exercise, we looked at \( \sqrt{3x + 7} > 2 \) and \( \sqrt{3x + 7} < 2 \). We solve these inequalities by first understanding how the square root expression behaves. After finding where the equation \( \sqrt{3x + 7} = 2 \) is true (at \( x = -1 \)), we check where the function is above or below this threshold.
This means \( \sqrt{3x + 7} > 2 \) is true for \( x > -1 \) and \( \sqrt{3x + 7} < 2 \) is true for \( x < -1 \). Inequalities can be visually understood on graphs, making them easier to solve.

Graphical Representation

Graphs provide a visual hint about how functions behave. When we graph a function, we can see precisely where it crosses certain lines or reaches particular values. This is extremely helpful when solving equations or inequalities.
For our function \( f(x) = \sqrt{3x + 7} \), graphing it on a coordinate plane shows us the shape and trend of the curve. The critical aspect to note here was the horizontal line \( y = 2 \), which represents the value we were comparing against.
In the graph,

• The point where \( f(x) = 2 \) is where the curve meets the line \( y = 2 \) at \( x = -1 \)
• Areas above the line (\( y > 2 \)) suggest \( \sqrt{3x+7} > 2 \)
• Regions below (\( y < 2 \)) mean \( \sqrt{3x+7} < 2 \)

Graphical representation not only offers a clear and quick understanding but also makes complex algebraic solutions much easier to verify.

Analytic Methods

Analytic methods involve using algebraic manipulation and logical reasoning to find exact solutions to equations or inequalities. Solving equations analytically requires us to isolate and find values for unknown variables.
For example, in the equation \( \sqrt{3x + 7} = 2 \), we squared both sides to remove the square root, leading to \( 3x + 7 = 4 \). Step by step, we isolated \( x \) by undoing operations:

• Subtract 7 from both sides: \( 3x = -3 \)
• Divide by 3: \( x = -1 \)

This systematic breakdown is the essence of analytic methods: solving by reducing the complexity of the problem until it becomes simple arithmetic.
Analytic methods are powerful because they deliver exact and precise answers, which graphical methods may only approximate visually.