Find The Value Of X Round To The Nearest Tenth

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

Finding the value of 'x' is a fundamental concept in mathematics, appearing across various branches like algebra, geometry, trigonometry, and calculus. Solving for 'x' often involves applying different techniques depending on the context of the problem. This comprehensive guide will explore various methods to find the value of 'x', focusing on how to round the answer to the nearest tenth when necessary. We will cover a range of examples, from simple linear equations to more complex scenarios involving quadratic equations and trigonometric functions. Mastering these techniques is crucial for success in many mathematical and scientific fields.

Understanding the Concept of Rounding to the Nearest Tenth

Before diving into the methods of solving for 'x', let's clarify the concept of rounding to the nearest tenth. The tenth place is the first digit after the decimal point. When rounding to the nearest tenth, we look at the digit in the hundredths place (the second digit after the decimal point).

• If the digit in the hundredths place is 5 or greater, we round the digit in the tenths place up by one.
• If the digit in the hundredths place is less than 5, we keep the digit in the tenths place as it is.

For example:

• 3.14 rounded to the nearest tenth is 3.1
• 3.15 rounded to the nearest tenth is 3.2
• 3.17 rounded to the nearest tenth is 3.2
• 3.12 rounded to the nearest tenth is 3.1

Solving for 'x' in Linear Equations

Linear equations are equations where the highest power of the variable 'x' is 1. Solving these equations typically involves isolating 'x' on one side of the equation using algebraic manipulations.

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 a whole number, so rounding is not necessary.

Example 2: Solve for x: 2x - 7 = 1.5

1. Add 7 to both sides: 2x = 8.5
2. Divide both sides by 2: x = 4.25

Rounding 4.25 to the nearest tenth gives us 4.3

Example 3: Solve for x: (x/4) + 2 = 6

1. Subtract 2 from both sides: x/4 = 4
2. Multiply both sides by 4: x = 16

Again, rounding isn't required here.

Solving for 'x' in Quadratic Equations

Quadratic equations are equations where the highest power of the variable 'x' is 2. These equations can be solved using various methods, including factoring, the quadratic formula, and completing the square.

Example 4: Solve for x: x² - 5x + 6 = 0

This equation can be factored as (x - 2)(x - 3) = 0. Therefore, the solutions are x = 2 and x = 3. No rounding is needed.

Example 5: Solve for x: x² + 4x - 1 = 0

This equation requires the quadratic formula: x = [-b ± √(b² - 4ac)] / 2a, where a = 1, b = 4, and c = -1.

Plugging in the values, we get: x = [-4 ± √(16 - 4(1)(-1))] / 2 = [-4 ± √20] / 2

This gives us two solutions: x ≈ 0.236 and x ≈ -4.236.

Rounding to the nearest tenth, we get x ≈ 0.2 and x ≈ -4.2.

Example 6: Solve for x: 2x² - 7x + 3 = 0

This quadratic equation can be solved by factoring: (2x - 1)(x - 3) = 0

This gives us two solutions: x = 1/2 = 0.5 and x = 3. No rounding needed for x=3, but x=0.5 is already to the nearest tenth.

Solving for 'x' in Trigonometric Equations

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

Example 7: Solve for x: sin(x) = 0.5

Using the inverse sine function (arcsin), we find x = arcsin(0.5) = 30° or π/6 radians. In this case, x is an exact value, no rounding is necessary. However, trigonometric functions often yield multiple solutions within a given range.

Example 8: Solve for x: cos(x) = 0.7

Using the inverse cosine function (arccos), we get x = arccos(0.7) ≈ 45.57°. Rounding to the nearest tenth, x ≈ 45.6°. Remember that the cosine function is positive in the first and fourth quadrants, leading to multiple solutions within a full 360° range.

Example 9: Solve for x: tan(x) = 2

Using the inverse tangent function (arctan), we find x = arctan(2) ≈ 63.43°. Rounding to the nearest tenth gives us x ≈ 63.4°. Like the cosine function, the tangent function also has multiple solutions within a given range.

Solving for 'x' in Exponential and Logarithmic Equations

Exponential equations involve variables in the exponent, while logarithmic equations involve logarithms. Solving these requires specific techniques.

Example 10: Solve for x: 2^x = 8

This can be rewritten as 2^x = 2³. Therefore, x = 3.

Example 11: Solve for x: log₂(x) = 3

This is equivalent to 2³ = x, so x = 8.

Example 12: Solve for x: e^x = 5 (where 'e' is the natural logarithm base)

Taking the natural logarithm of both sides, we get x = ln(5) ≈ 1.609. Rounding to the nearest tenth gives x ≈ 1.6.

Solving Systems of Equations for 'x'

Sometimes, 'x' needs to be solved within a system of equations. Methods like substitution or elimination can be used.

Example 13: Solve for x:

x + y = 5 x - y = 1

Adding the two equations, we get 2x = 6, so x = 3.

Example 14: Solve for x:

2x + y = 7 x - 2y = 4

We can solve this system using substitution or elimination. Using elimination, multiply the second equation by 2: 2x - 4y = 8. Subtract this from the first equation to eliminate y: (2x + y) - (2x - 4y) = 7 - 8, which simplifies to 5y = -1, so y = -0.2. Substituting this back into the first equation, we get 2x - 0.2 = 7, so 2x = 7.2 and x = 3.6.

Importance of Checking Your Solutions

After solving for 'x', it's crucial to check your answer by substituting it back into the original equation(s). This helps ensure accuracy and identifies any potential errors in your calculations.

Conclusion: Mastering the Art of Finding 'x'

Finding the value of 'x' is a core skill in mathematics. This guide has covered various methods for solving for 'x' in different types of equations, emphasizing the importance of rounding to the nearest tenth when necessary. By understanding these techniques and practicing regularly, you can build a strong foundation in algebra and other related mathematical fields. Remember to always check your solutions to ensure accuracy and develop a thorough understanding of the underlying mathematical principles. Consistent practice and attention to detail will greatly enhance your problem-solving skills.