Problem 82 - College Algebra with Modeling and Visualization

Question

Solve the inequality. Approximate the endpoints to the nearest thousandth when appropriate. $$ 5.1 x-\pi \geq \sqrt{3}-1.7 x $$

Step-by-Step Solution

Answer

x ≥ 0.717.

Step 1: Move Variables to One Side

Start by moving all terms involving the variable \(x\) to one side of the inequality. Add \(1.7x\) to both sides:\[5.1x + 1.7x - \,\pi \,\geq \,\sqrt{3}.\]

Step 2: Combine Like Terms

Now, combine the terms involving \(x\).\[ (5.1 + 1.7)x \,\geq \,\pi \, + \,\sqrt{3}\]This simplifies to:\[ 6.8x \,\geq \,\pi + \,\sqrt{3}\]

Step 3: Solve for x

Solve the inequality for \(x\) by dividing both sides by 6.8:\[ x \,\geq \,\frac{\pi \, + \,\sqrt{3}}{6.8}\].

Step 4: Calculate the Numerical Result

Using a calculator, approximate the values of \(\pi\) and \(\sqrt{3}\) and compute the expression:\[ \frac{3.142 \, + \,1.732}{6.8} = \frac{4.874}{6.8} \approx 0.717\]Thus, \(x \geq 0.717\).

Key Concepts

Algebra

Algebra is a fundamental branch of mathematics that deals with symbols and the rules for manipulating these symbols. In this exercise, we are using algebraic techniques to manipulate and solve an inequality involving the variable \( x \).

When dealing with inequalities, the approach to solving them is similar to that of solving equations. However, inequalities require extra attention because they express a range of values rather than a single result. The first step in solving the given inequality was moving terms involving \( x \) to one side, which simplifies the process by isolating \( x \) on one side of the inequality symbol.

• Addition or subtraction of the same number or variable on both sides of an inequality preserves the inequality.
• Combining like terms allows you to simplify expressions, making it easy to focus on the core of the inequality.

Using these basic algebraic tools gives clarity and direction when aiming to isolate the variable and solve the inequality.

Mathematical Inequalities

Mathematical inequalities are statements about the relative size or order of two values. Understanding inequalities is crucial, especially as they represent conditions of "greater than," "less than," or "equal to." In this problem, our inequality was \(5.1x - \pi \geq \sqrt{3} - 1.7x\).

Upon rearranging and simplifying this inequality, we sought to find the range of values for \( x \) that make the inequality hold true:

• By moving all terms involving \( x \) to one side, you can isolate the variable of interest.
• Make sure to combine like terms, simplifying the expression to see the inequality's structure clearly.

Finally, the inequality was solved by dividing both sides by a constant, which is valid as long as the constant is positive, preserving the direction of the inequality. Once we solved for \( x \), we found \( x \geq 0.717\), confirming the range of possible solutions.

Approximation Techniques

Approximation techniques are useful when you're working with irrational numbers in your calculations. Numbers like \( \pi \) and \( \sqrt{3} \), which appear in our inequality, are known to be irrational because their exact forms cannot be fully expressed as finite decimals.

To approximate these values to the nearest thousandth:

• \( \pi \) is approximately 3.142.
• \( \sqrt{3} \) is approximately 1.732.

Once you have these approximate values, you can perform your division more easily. In this exercise, this step was essential for getting a simple, actionable numerical solution for \( x \). By calculating:

\[ \frac{3.142 + 1.732}{6.8} \approx 0.717 \],
we arrive at our answer, \( x \geq 0.717 \), now expressed in a practical form for further use or interpretation.