Find The Value Of X Rounded To The Nearest Tenth

Find the Value of x Rounded to the Nearest Tenth: A Comprehensive Guide

Finding the value of 'x' and rounding to the nearest tenth is a fundamental skill in mathematics, applicable across various fields, from basic algebra to advanced calculus. This comprehensive guide will explore different methods and scenarios for solving for 'x' and accurately rounding the solution. We'll cover a range of equations, including linear equations, quadratic equations, trigonometric equations, and more, providing step-by-step explanations and examples. Mastering this skill is crucial for students and professionals alike, ensuring accuracy and efficiency in problem-solving.

Understanding Rounding to the Nearest Tenth

Before diving into solving for 'x', let's solidify our understanding of rounding to the nearest tenth. The tenth place is the first digit after the decimal point. To round to the nearest tenth, we look at the digit in the hundredths place (the second digit after the decimal point).

• If the hundredths digit is 5 or greater (5, 6, 7, 8, 9), we round the tenths digit up. For example, 3.15 rounds up to 3.2, and 7.86 rounds up to 7.9.

• If the hundredths digit is less than 5 (0, 1, 2, 3, 4), we keep the tenths digit the same. For example, 2.43 rounds down to 2.4, and 9.92 rounds down to 9.9.

Solving for 'x' in Linear Equations

Linear equations are equations of the form ax + b = c, where a, b, and c are constants, and x is the variable we need to solve for. The process involves isolating 'x' using algebraic manipulation.

Example 1: Solve for x: 3x + 5 = 14

1. Subtract 5 from both sides: 3x = 9
2. Divide both sides by 3: x = 3

In this case, x is already a whole number, so rounding to the nearest tenth isn't necessary. The solution is simply x = 3.0

Example 2: Solve for x: 2.5x - 7 = 10.5

1. Add 7 to both sides: 2.5x = 17.5
2. Divide both sides by 2.5: x = 7

Again, x is a whole number, so the solution, rounded to the nearest tenth, is x = 7.0.

Example 3: Solve for x: 0.8x + 2.3 = 5.9

1. Subtract 2.3 from both sides: 0.8x = 3.6
2. Divide both sides by 0.8: x = 4.5

Here, x is already expressed to the nearest tenth, so no rounding is required. The solution is x = 4.5.

Solving for 'x' in Quadratic Equations

Quadratic equations are equations of the form ax² + bx + c = 0, where a, b, and c are constants, and x is the variable. Solving quadratic equations typically involves the quadratic formula:

x = [-b ± √(b² - 4ac)] / 2a

Example 4: Solve for x: x² + 3x - 10 = 0

Here, a = 1, b = 3, and c = -10. Substituting these values into the quadratic formula gives:

x = [-3 ± √(3² - 4 * 1 * -10)] / (2 * 1) = [-3 ± √49] / 2 = [-3 ± 7] / 2

This yields two solutions: x = 2 and x = -5. Both are whole numbers, so rounded to the nearest tenth, they are 2.0 and -5.0.

Example 5: Solve for x: 2x² - 5x - 3 = 0

Using the quadratic formula with a = 2, b = -5, and c = -3:

x = [5 ± √((-5)² - 4 * 2 * -3)] / (2 * 2) = [5 ± √49] / 4 = [5 ± 7] / 4

This gives two solutions: x = 3 and x = -0.5. Both are already expressed to the nearest tenth.

Solving for 'x' in Trigonometric Equations

Trigonometric equations involve trigonometric functions like sine, cosine, and tangent. Solving these often requires using inverse trigonometric functions and understanding the unit circle.

Example 6: Solve for x: sin(x) = 0.5

Using the inverse sine function (arcsin or sin⁻¹):

x = arcsin(0.5)

The principal value is x = 30° (or π/6 radians). However, sine is positive in both the first and second quadrants, so there's another solution in the second quadrant: x = 150° (or 5π/6 radians). These are exact values. If we require decimal approximations rounded to the nearest tenth, these values remain unchanged.

Example 7: Solve for x: cos(x) = -0.7

x = arccos(-0.7)

This results in an angle in the second and third quadrants. Using a calculator, we get an approximate value for the principal angle, and then find the other solution using the symmetry of the cosine function. The solutions, rounded to the nearest tenth, might be something like x ≈ 134.4° and x ≈ 225.6°. (Note: the precise values depend on the calculator used).

Solving for 'x' in Exponential and Logarithmic Equations

Exponential equations involve variables in the exponent, while logarithmic equations involve logarithms. Solving these often requires using logarithmic properties or exponential properties.

Example 8: Solve for x: 2ˣ = 10

Taking the logarithm of both sides (base 10 or natural logarithm):

x log(2) = log(10) => x = log(10) / log(2) ≈ 3.32

Rounded to the nearest tenth, x ≈ 3.3

Example 9: Solve for x: log₂(x) = 3

This means 2³ = x, so x = 8. This is a whole number; rounded to the nearest tenth it is 8.0.

Handling More Complex Equations

Many equations involve multiple steps and various algebraic techniques. The key is to carefully isolate 'x' using the appropriate methods for each type of equation. This often involves a combination of techniques like factoring, completing the square, using the quadratic formula, applying trigonometric identities, or utilizing logarithmic and exponential properties. Remember to always check your solution by substituting it back into the original equation.

Importance of Accuracy and Precision

Accuracy and precision are vital, especially in fields like engineering, physics, and finance. Incorrect rounding can lead to significant errors in calculations and final results. Always pay close attention to the required level of precision, indicated by the number of decimal places or significant figures.

Conclusion: Mastering 'x'

Finding the value of 'x' and rounding to the nearest tenth requires a thorough understanding of algebraic manipulation, equation solving techniques, and rounding principles. This comprehensive guide has covered various equation types, from simple linear equations to more complex trigonometric and logarithmic equations. By mastering these techniques, you'll enhance your mathematical problem-solving skills and increase accuracy in your calculations across various disciplines. Remember, practice is key to perfecting these skills, so work through numerous examples to build confidence and proficiency. The ability to accurately solve for 'x' and round to the nearest tenth is a valuable asset in numerous academic and professional contexts.