Find X To The Nearest Tenth

Finding 'x' to the Nearest Tenth: A Comprehensive Guide

Finding the value of 'x' to the nearest tenth is a common task in mathematics, particularly in algebra, trigonometry, and calculus. This process often involves solving equations, using formulas, or applying various numerical methods. This comprehensive guide explores different scenarios where finding 'x' to the nearest tenth is necessary, providing step-by-step solutions and practical examples. We'll cover various approaches, from simple linear equations to more complex trigonometric and logarithmic problems.

Understanding the Concept of "Nearest Tenth"

Before delving into different methods, let's clarify what "nearest tenth" means. It refers to rounding a number to one decimal place. 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, we round the tenths digit up (increase it by 1).
• If the hundredths digit is less than 5, we keep the tenths digit as it is.

For example:

• 3.14 rounds to 3.1
• 2.78 rounds to 2.8
• 5.65 rounds to 5.7
• 9.99 rounds to 10.0

Solving Linear Equations to Find x

Linear equations are the simplest type of equations to solve for 'x'. They involve only one variable raised to the power of 1. The general form is ax + b = c, where 'a', 'b', and 'c' are constants.

Example 1: Solve 2x + 5 = 11 for x to the nearest tenth.

1. Subtract 5 from both sides: 2x = 6
2. Divide both sides by 2: x = 3

In this case, x is already a whole number, so rounding to the nearest tenth is unnecessary; x = 3.0

Example 2: Solve 3x - 7 = 1.4 for x to the nearest tenth.

1. Add 7 to both sides: 3x = 8.4
2. Divide both sides by 3: x = 2.8

Here, x = 2.8, which is already expressed to the nearest tenth.

Solving Quadratic Equations to Find x

Quadratic equations have a variable raised to the power of 2. The general form is ax² + bx + c = 0. Solving quadratic equations often involves using the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

Example 3: Solve x² + 4x - 5 = 0 for x to the nearest tenth.

Here, a = 1, b = 4, and c = -5. Substituting these values into the quadratic formula:

x = (-4 ± √(4² - 4 * 1 * -5)) / (2 * 1) x = (-4 ± √36) / 2 x = (-4 ± 6) / 2

This gives two solutions:

x = (-4 + 6) / 2 = 1 x = (-4 - 6) / 2 = -5

Both solutions are whole numbers; therefore, to the nearest tenth: x = 1.0 and x = -5.0

Example 4: Solve 2x² - 3x - 2 = 0 for x to the nearest tenth.

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

x = (3 ± √((-3)² - 4 * 2 * -2)) / (2 * 2) x = (3 ± √25) / 4 x = (3 ± 5) / 4

This gives two solutions:

x = (3 + 5) / 4 = 2 x = (3 - 5) / 4 = -0.5

The solutions are x = 2.0 and x = -0.5

Using Trigonometry to Find x

Trigonometry involves relationships between angles and sides of triangles. Finding 'x' often requires applying trigonometric functions like sine, cosine, and tangent.

Example 5: In a right-angled triangle, the hypotenuse is 10 cm and the opposite side to an angle of 30° is 'x'. Find x to the nearest tenth.

We use the sine function: sin(30°) = opposite / hypotenuse = x / 10

Solving for x: x = 10 * sin(30°) = 10 * 0.5 = 5 cm

Therefore, x = 5.0 cm

Example 6: In a right-angled triangle, the adjacent side to an angle of 45° is 7 cm and the hypotenuse is 'x'. Find x to the nearest tenth.

We use the cosine function: cos(45°) = adjacent / hypotenuse = 7 / x

Solving for x: x = 7 / cos(45°) = 7 / 0.707 ≈ 9.899 ≈ 9.9 cm

Therefore, x ≈ 9.9 cm

Solving Logarithmic Equations to Find x

Logarithmic equations involve logarithms. Solving for 'x' might require using logarithmic properties and changing the equation to an exponential form.

Example 7: Solve log₁₀(x) = 2 for x.

This means 10² = x, so x = 100. To the nearest tenth: x = 100.0

Example 8: Solve ln(x) = 1.5 for x (ln denotes the natural logarithm, base e).

This means e¹.⁵ = x. Using a calculator: x ≈ 4.481 ≈ 4.5

Numerical Methods for Finding x

For complex equations that cannot be solved algebraically, numerical methods are used to approximate the value of 'x'. These methods involve iterative processes to refine an initial guess until a desired level of accuracy is achieved. Examples include the Newton-Raphson method and the bisection method. These methods are often implemented using computational tools.

Importance of Accuracy and Significant Figures

When working with measurements or experimental data, the accuracy of the answer is crucial. Rounding to the nearest tenth may not always be appropriate. The number of significant figures should reflect the precision of the input data. If the input values have only one significant figure, reporting 'x' to several decimal places would be misleading.

Conclusion

Finding 'x' to the nearest tenth involves a range of mathematical techniques, from straightforward algebraic manipulations to more complex methods involving trigonometry, logarithms, and numerical analysis. Understanding the underlying concepts and choosing the appropriate method are key to solving problems accurately and efficiently. Always consider the context of the problem and the required level of precision when rounding your answer. Remember to check your work to ensure the solution is reasonable and consistent with the problem's parameters. Practice is vital to mastering these techniques and building confidence in tackling more complex mathematical challenges.