Problem 81 - College Algebra with Modeling and Visualization

Question

Solve the inequality. Approximate the endpoints to the nearest thousandth when appropriate. $$ \pi x-5.12 \leq \sqrt{2} x-5.7(x-1.1) $$

Step-by-Step Solution

Answer

\( x \leq 1.533 \).

Step 1: Simplify the inequality

Let's first expand the right-hand side of the inequality. The expression is \(\sqrt{2}x - 5.7(x-1.1)\). Distribute \(-5.7\): \(-5.7(x-1.1) = -5.7x + 5.7 \cdot 1.1\)This simplifies to:\(-5.7x + 6.27\).Now, the inequality is: \( \pi x - 5.12 \leq \sqrt{2}x - 5.7x + 6.27 \).

Step 2: Combine like terms

We need to combine like terms from both sides of the inequality.The inequality is:\(\pi x - 5.12 \leq \sqrt{2}x - 5.7x + 6.27\).Combine the terms on the right:\(\sqrt{2}x - 5.7x = (\sqrt{2} - 5.7)x\).So the inequality becomes:\(\pi x - 5.12 \leq (\sqrt{2} - 5.7)x + 6.27\).

Step 3: Move all x terms to one side

Subtract \((\sqrt{2} - 5.7)x\) from both sides of the inequality:\(\pi x - (\sqrt{2} - 5.7)x - 5.12 \leq 6.27\).This simplifies to:\((\pi - \sqrt{2} + 5.7)x - 5.12 \leq 6.27\).

Step 4: Solve for x

Now let's solve the inequality for \(x\).First, add 5.12 to both sides:\((\pi - \sqrt{2} + 5.7)x \leq 11.39\).Now, divide both sides by \((\pi - \sqrt{2} + 5.7)\):\[ x \leq \frac{11.39}{\pi - \sqrt{2} + 5.7} \].

Step 5: Calculate numerical solution

Substitute the approximate values for \(\pi\) and \(\sqrt{2}\). Use \(\pi \approx 3.142\) and \(\sqrt{2} \approx 1.414\):\( \pi - \sqrt{2} + 5.7 \approx 3.142 - 1.414 + 5.7 = 7.428\).Therefore:\[ x \leq \frac{11.39}{7.428} \approx 1.533 \].Round the result to the nearest thousandth:

Step 6: Final Answer and rounding

Based on the calculation, the final answer rounded to the nearest thousandth is:\( x \leq 1.533 \).

Key Concepts

Algebraic Manipulation

Algebraic manipulation is an essential skill in solving equations and inequalities, allowing us to transform complex expressions into simpler, more manageable forms. In this specific problem, we start by dealing with an inequality \[ \pi x - 5.12 \leq \sqrt{2} x - 5.7(x - 1.1) \].
First, distribute the \(-5.7\) on the right side: - This involves multiplying each term inside the parenthesis by \(-5.7\).
Next, simplify it to \(-5.7x + 6.27\).
Our goal in algebraic manipulation is to isolate terms, particularly when dealing with inequalities:

• Combine like terms
• Move \(x\)-terms to one side
• Isolate constants on the other side

This involves organizing all similar terms on each side of the inequality. For instance, combining terms \(\sqrt{2}x\) and \(-5.7x\) results in \((\sqrt{2} - 5.7)x\). This way, you can then more effectively solve for \(x\) by operating simplifications and transpositions step-by-step.

Approximation

Approximation is a key concept in mathematics, particularly when working with irrational numbers like \(\pi\) and \(\sqrt{2}\).
In many problems, we replace these values with numerical approximations to facilitate easier calculations:

• \(\pi \approx 3.142\)
• \(\sqrt{2} \approx 1.414\)

Using approximations helps when complex expressions resist simplification analytically. In this exercise, since the inequality's endpoints needed to be approximated to the nearest thousandth, substituting these values simplifies our final computations.
Approximation allows us to work with a cleaner, finite number of decimals, thus avoiding dealing with cumbersome irrational values. This is crucial when aiming to understand roughly how numbers relate via simpler calculations, which demonstrate how close our solutions can be to their exact counterparts in practical scenarios.

Numerical Solution

Numerical solutions are a handy way of finding values of variables through actual numerical computation rather than through algebraic manipulations alone. In the case of our inequality problem, after simplifying expressions, our task is to solve \[ x \leq \frac{11.39}{\pi - \sqrt{2} + 5.7} \].
Here, we implement the approximate values discussed earlier. Substituting \(\pi\) with \(3.142\) and \(\sqrt{2}\) with \(1.414\), the denominator becomes \((7.428)\).
Now, performing the division: \[ x \leq \frac{11.39}{7.428} \approx 1.533 \].
This numerical answer, rounded to the nearest thousandth, provides the endpoint of the solution set for \(x\). Rounding emphasizes finding a balance between precision and practical understanding. This final number represents how close the estimated mathematical solution can be under specified approximation limits.