Directions Solve For X. Round To The Nearest Tenth

Solving for x is a fundamental concept in algebra, serving as the cornerstone for more advanced mathematical topics. Mastering this skill allows you to tackle various equations and problems, making it an essential tool in both academic and real-world scenarios. This guide will walk you through the process of solving for x, providing clear steps and examples, while rounding the answers to the nearest tenth when necessary.

Understanding the Basics

At its core, solving for x involves isolating the variable x on one side of an equation. This means manipulating the equation using mathematical operations to get x by itself. The key principle is to maintain the equation's balance by performing the same operation on both sides.

Key Concepts:

• Equation: A mathematical statement that asserts the equality of two expressions.
• Variable: A symbol (usually a letter like x, y, or z) representing an unknown value.
• Coefficient: A number multiplied by a variable (e.g., in 3x, 3 is the coefficient).
• Constant: A fixed value in an equation (e.g., 5 in x + 5 = 9).
• Inverse Operations: Operations that undo each other (e.g., addition and subtraction, multiplication and division).

Steps to Solve for x

Here’s a step-by-step approach to solving for x in various types of equations:

1. Simplify Both Sides of the Equation

Before isolating x, it’s crucial to simplify each side of the equation by:

• Combining Like Terms: Combine terms with the same variable and exponent. For example, in 3x + 2x - 5 = 10, combine 3x and 2x to get 5x - 5 = 10.
• Distributing: If there are parentheses, distribute any coefficients. For example, in 2(x + 3) = 12, distribute the 2 to get 2x + 6 = 12.

2. Isolate the Term with x**

Move all terms that do not contain x to the opposite side of the equation. Use inverse operations to do this:

• If a constant is added to the term with x, subtract it from both sides. For example, in x + 5 = 9, subtract 5 from both sides to get x = 4.
• If a constant is subtracted from the term with x, add it to both sides. For example, in x - 3 = 7, add 3 to both sides to get x = 10.

3. Solve for x**

Once the term with x is isolated, solve for x by:

• If x is multiplied by a coefficient, divide both sides by the coefficient. For example, in 3x = 15, divide both sides by 3 to get x = 5.
• If x is divided by a number, multiply both sides by that number. For example, in x / 2 = 6, multiply both sides by 2 to get x = 12.

4. Check Your Solution

After finding a value for x, plug it back into the original equation to verify that it makes the equation true. This step ensures that your solution is correct.

Examples of Solving for x

Let's walk through several examples to illustrate these steps.

Example 1: Basic Linear Equation

Solve for x: 2x + 3 = 9

1. Simplify: The equation is already simplified.
2. Isolate the term with x: Subtract 3 from both sides:
   • 2x + 3 - 3 = 9 - 3
   • 2x = 6
3. Solve for x: Divide both sides by 2:
   • 2x / 2 = 6 / 2
   • x = 3
4. Check: Plug x = 3 back into the original equation:
   • 2(3) + 3 = 9
   • 6 + 3 = 9
   • 9 = 9 (The solution is correct.)

Example 2: Equation with Distribution

Solve for x: 3(x - 2) = 15

1. Simplify: Distribute the 3:
   • 3x - 6 = 15
2. Isolate the term with x: Add 6 to both sides:
   • 3x - 6 + 6 = 15 + 6
   • 3x = 21
3. Solve for x: Divide both sides by 3:
   • 3x / 3 = 21 / 3
   • x = 7
4. Check: Plug x = 7 back into the original equation:
   • 3(7 - 2) = 15
   • 3(5) = 15
   • 15 = 15 (The solution is correct.)

Example 3: Equation with Fractions

Solve for x: x / 4 + 2 = 5

1. Simplify: The equation is already simplified.
2. Isolate the term with x: Subtract 2 from both sides:
   • x / 4 + 2 - 2 = 5 - 2
   • x / 4 = 3
3. Solve for x: Multiply both sides by 4:
   • (x / 4) * 4 = 3 * 4
   • x = 12
4. Check: Plug x = 12 back into the original equation:
   • 12 / 4 + 2 = 5
   • 3 + 2 = 5
   • 5 = 5 (The solution is correct.)

Example 4: Equation with Like Terms

Solve for x: 5x - 2x + 7 = 16

1. Simplify: Combine like terms:
   • 3x + 7 = 16
2. Isolate the term with x: Subtract 7 from both sides:
   • 3x + 7 - 7 = 16 - 7
   • 3x = 9
3. Solve for x: Divide both sides by 3:
   • 3x / 3 = 9 / 3
   • x = 3
4. Check: Plug x = 3 back into the original equation:
   • 5(3) - 2(3) + 7 = 16
   • 15 - 6 + 7 = 16
   • 16 = 16 (The solution is correct.)

Example 5: Equation with Decimal and Rounding

Solve for x: 4.2x - 1.5 = 10.3 (Round to the nearest tenth)

1. Simplify: The equation is already simplified.
2. Isolate the term with x: Add 1.5 to both sides:
   • 4.2x - 1.5 + 1.5 = 10.3 + 1.5
   • 4.2x = 11.8
3. Solve for x: Divide both sides by 4.2:
   • 4.2x / 4.2 = 11.8 / 4.2
   • x ≈ 2.8095
4. Round to the nearest tenth:
   • x ≈ 2.8
5. Check: Plug x = 2.8 back into the original equation:
   • 4.2(2.8) - 1.5 = 10.3
   • 11.76 - 1.5 = 10.3
   • 10.26 ≈ 10.3 (The solution is approximately correct due to rounding.)

Example 6: Equation with Negative Numbers

Solve for x: -3x + 5 = -4

1. Simplify: The equation is already simplified.
2. Isolate the term with x: Subtract 5 from both sides:
   • -3x + 5 - 5 = -4 - 5
   • -3x = -9
3. Solve for x: Divide both sides by -3:
   • -3x / -3 = -9 / -3
   • x = 3
4. Check: Plug x = 3 back into the original equation:
   • -3(3) + 5 = -4
   • -9 + 5 = -4
   • -4 = -4 (The solution is correct.)

Example 7: Multi-Step Equation

Solve for x: 2(x + 3) - 4 = 8

1. Simplify: Distribute the 2 and combine like terms:
   • 2x + 6 - 4 = 8
   • 2x + 2 = 8
2. Isolate the term with x: Subtract 2 from both sides:
   • 2x + 2 - 2 = 8 - 2
   • 2x = 6
3. Solve for x: Divide both sides by 2:
   • 2x / 2 = 6 / 2
   • x = 3
4. Check: Plug x = 3 back into the original equation:
   • 2(3 + 3) - 4 = 8
   • 2(6) - 4 = 8
   • 12 - 4 = 8
   • 8 = 8 (The solution is correct.)

Example 8: Solving for x with Parentheses and Decimals

Solve for x: 1.5(x - 2.5) + 3.2 = 8.9 (Round to the nearest tenth)

1. Simplify: Distribute the 1.5:
   • 1. 5x - 3.75 + 3.2 = 8.9
   • 1. 5x - 0.55 = 8.9
2. Isolate the term with x: Add 0.55 to both sides:
   • 1. 5x - 0.55 + 0.55 = 8.9 + 0.55
   • 1. 5x = 9.45
3. Solve for x: Divide both sides by 1.5:
   • 1. 5x / 1.5 = 9.45 / 1.5
   • x ≈ 6.3
4. Check: Plug x = 6.3 back into the original equation:
   • 1. 5(6.3 - 2.5) + 3.2 = 8.9
   • 1. 5(3.8) + 3.2 = 8.9
   • 5. 7 + 3.2 = 8.9
   • 8. 9 = 8.9 (The solution is correct.)

Example 9: Solving for x in a More Complex Equation

Solve for x: 4(2x - 1) + 3 = 2(3x + 1) - 5

1. Simplify: Distribute and combine like terms:
   • 8x - 4 + 3 = 6x + 2 - 5
   • 8x - 1 = 6x - 3
2. Isolate the term with x: Subtract 6x from both sides:
   • 8x - 1 - 6x = 6x - 3 - 6x
   • 2x - 1 = -3
3. Add 1 to both sides:
   • 2x - 1 + 1 = -3 + 1
   • 2x = -2
4. Solve for x: Divide both sides by 2:
   • 2x / 2 = -2 / 2
   • x = -1
5. Check: Plug x = -1 back into the original equation:
   • 4(2(-1) - 1) + 3 = 2(3(-1) + 1) - 5
   • 4(-2 - 1) + 3 = 2(-3 + 1) - 5
   • 4(-3) + 3 = 2(-2) - 5
   • -12 + 3 = -4 - 5
   • -9 = -9 (The solution is correct.)

Example 10: Solving for x with Radicals (and Rounding)

Solve for x: x^2 = 10 (Round to the nearest tenth)

1. Simplify: The equation is already simplified.
2. Solve for x: Take the square root of both sides:
   • √(x^2) = ±√10
   • x = ±√10
3. Calculate the square root of 10:
   • x ≈ ±3.162277...
4. Round to the nearest tenth:
   • x ≈ ±3.2

Therefore, the solutions are x ≈ 3.2 and x ≈ -3.2.

Common Mistakes to Avoid

• Not Distributing Properly: Ensure you distribute across all terms inside parentheses.
• Incorrectly Combining Like Terms: Only combine terms with the same variable and exponent.
• Forgetting to Perform the Same Operation on Both Sides: This maintains the equation's balance.
• Not Checking Your Solution: Always plug the solution back into the original equation to verify its correctness.
• Sign Errors: Pay close attention to signs when adding, subtracting, multiplying, or dividing negative numbers.
• Rounding Too Early: Wait until the final step to round your answer to maintain accuracy. Rounding intermediate values can lead to significant errors in the final result.
• Ignoring the Order of Operations: Always follow the correct order of operations (PEMDAS/BODMAS) when simplifying expressions.

Advanced Techniques

Quadratic Equations

Quadratic equations are in the form ax² + bx + c = 0. To solve them, you can use:

• Factoring: Break down the quadratic expression into two binomials.
• Quadratic Formula: x = (-b ± √(b² - 4ac)) / (2a)
• Completing the Square: Transform the equation into a perfect square trinomial.

Systems of Equations

Systems of equations involve two or more equations with the same variables. To solve them, you can use:

• Substitution: Solve one equation for one variable and substitute it into the other equation.
• Elimination: Add or subtract the equations to eliminate one variable.
• Graphing: Plot the equations on a graph and find the point of intersection.

Practical Applications

Solving for x isn't just an abstract mathematical exercise; it has numerous practical applications in various fields:

• Finance: Calculating interest rates, loan payments, and investment returns.
• Physics: Determining velocity, acceleration, and force in mechanics problems.
• Engineering: Designing structures, circuits, and systems.
• Computer Science: Developing algorithms, optimizing code, and solving problems in artificial intelligence.
• Everyday Life: Calculating discounts, determining proportions in cooking, and managing personal finances.

Conclusion

Solving for x is a crucial skill that forms the foundation of algebra and has wide-ranging applications in various fields. By understanding the basic principles, following the step-by-step approach, and practicing regularly, you can master this skill and confidently tackle more complex mathematical problems. Remember to always simplify, isolate, solve, and check your solutions to ensure accuracy, and pay close attention to rounding instructions when necessary. With consistent effort and a solid understanding of the underlying concepts, solving for x will become second nature, empowering you to excel in mathematics and beyond.