Problem 33 - A Graphical Approach to College Algebra

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) \(x > -1\); (c) \(x < -1\).

Step 1: Isolate the Square Root

Our given equation is \( \sqrt{3x+7} = 2 \). Since the square root is already isolated, we can proceed to the next step.

Step 2: Square Both Sides

To eliminate the square root, square both sides of the equation: \((\sqrt{3x+7})^2 = 2^2\). This simplifies to \(3x + 7 = 4\).

Step 3: Solve for 'x'

Rearrange the equation to solve for \(x\):Subtract 7 from both sides: \(3x = 4 - 7\), which simplifies to \(3x = -3\).Divide by 3: \(x = -1\).

Step 4: Verify the Solution

Substitute \(x = -1\) back into the original equation to verify: \(\sqrt{3(-1) + 7} = \sqrt{4} = 2\). This confirms our solution is correct.

Step 5: Graph the Function

Graph the function \(y = \sqrt{3x + 7}\) to visualize the solution. The point where the graph intersects \(y = 2\) confirms \(x = -1\) as the solution.

Step 6: Solve Inequality (b)

For \(\sqrt{3x+7} > 2\), observe the graph and calculate:\(3x + 7 > 4\) (by squaring both sides again), which implies \(3x > -3\). Therefore, \(x > -1\). According to the graph, this is correct as the function is increasing beyond \(x = -1\).

Step 7: Solve Inequality (c)

For \(\sqrt{3x+7} < 2\), solve similarly:\(3x + 7 < 4\), giving \(3x < -3\). This simplifies to \(x < -1\). The graph confirms this, as the function remains below \(y = 2\) for \(x < -1\).

Key Concepts

Solving Equations

When solving equations in algebra, finding the value of the unknown variable is the main goal. In our exercise, we are given the equation \( \sqrt{3x+7} = 2 \). This involves a square root, which is often a bit tricky to handle.
To solve, we must "undo" the square root. We achieve this by squaring both sides of the equation, which removes the square root:

• Originally: \( \sqrt{3x + 7} = 2 \)
• After squaring: \( 3x + 7 = 4 \)

Next, we solve for the variable \( x \) by isolating it:

• Subtract 7 from both sides: \( 3x = 4 - 7 = -3 \)
• Divide by 3: \( x = -1 \)

This reveals \( x = -1 \) as the solution. To be sure it's correct, plug \( x = -1 \) back into the original equation and check if both sides are equal. They are, so \( x = -1 \) is indeed the correct answer.

Inequalities

Inequalities involve finding the set of values that satisfy a condition. They use symbols like \( > \) (greater than) and \( < \) (less than). In our context, we tackle two inequalities:- For \( \sqrt{3x + 7} > 2 \), after isolating the square root and squaring, we find the inequality becomes \( 3x + 7 > 4 \). Solving this gives \( x > -1 \). It tells us for any \( x \) greater than \(-1\), the original inequality holds true.- Conversely, for \( \sqrt{3x + 7} < 2 \), it transforms into \( 3x + 7 < 4 \), leading to \( x < -1 \). This means that the inequality holds for all \( x \) less than \(-1\).Inequalities tell us about ranges of values rather than one specific solution. Checking against the graph ensures our algebraic solutions make practical sense.

Graphical Solution

Graphical solutions provide a visual aid to solving equations and inequalities. For the equation \( \sqrt{3x+7}=2 \), plotting helps see where the function \( y = \sqrt{3x+7} \) meets \( y = 2 \).
The intersection point on the graph confirms our solution \( x = -1 \). It is the point where both equations share the same value for \( y \). When reading graphs:

• The \( y \)-axis shows the value of \( \sqrt{3x + 7} \)
• The \( x \)-coordinate of the intersection shows our solution for \( x \)

For inequalities, observing the graph tells us where \( \sqrt{3x + 7} \) is above or below \( y = 2 \). This confirms the solution to (b) and (c). Graphs are great for verifying and visualizing algebraic work, making the math more intuitive.

Square Root

The square root function \( \sqrt{x} \) is important in solving our given problems. It transforms a number into a value that, when multiplied by itself, results in the original number.
In our equation, \( \sqrt{3x + 7} \) represents the output of a process affecting \( x \):

• The expression \( 3x + 7 \) is inside the square root
• Solving our equation rests on "undoing" the square root by squaring both sides

Understanding how square roots and their inverses (squaring) work is key. By squaring the square root, you open up the expression, making the real work (solving for \( x \)) straightforward as the square root itself is removed.