Solving Equations And Inequalities A Step By Step Guide

Jul 18, 2025 by ADMIN 56 views

Solving Equations and Inequalities A Comprehensive Guide

Hey guys! Welcome to this comprehensive guide on solving equations and inequalities. In this article, we'll dive into the nitty-gritty of finding the value of x in various mathematical problems. We'll tackle everything from basic algebraic equations to inequalities, ensuring you have a solid understanding of the fundamental concepts and techniques. So, grab your pencils, and let’s get started!

1. Solving the Equation: (x+1)/(x-2) = 3

In this section, we're going to solve the equation (x+1)/(x-2) = 3. This is a classic algebraic problem that requires us to isolate x. The key here is to understand how to manipulate the equation while maintaining its balance. We'll walk through each step carefully, so you can see how we arrive at the solution.

Step-by-Step Solution

First things first, to get rid of the fraction, we need to multiply both sides of the equation by (x-2). This is a crucial step because it simplifies the equation and makes it easier to work with. So, let's do that:

(x+1)/(x-2) * (x-2) = 3 * (x-2)

This simplifies to:

x + 1 = 3(x - 2)

Now, we need to distribute the 3 on the right side of the equation. This means multiplying both the x and the -2 by 3:

x + 1 = 3x - 6

Next, we want to get all the x terms on one side and the constants on the other. Let's subtract x from both sides:

x - x + 1 = 3x - x - 6

This simplifies to:

1 = 2x - 6

Now, let’s add 6 to both sides to isolate the term with x:

1 + 6 = 2x - 6 + 6

Which simplifies to:

7 = 2x

Finally, to solve for x, we divide both sides by 2:

7/2 = 2x/2

So, the solution is:

x = 7/2

Checking the Solution

It’s always a good idea to check your solution to make sure it’s correct. Plug x = 7/2 back into the original equation:

(7/2 + 1) / (7/2 - 2) = 3

(7/2 + 2/2) / (7/2 - 4/2) = 3

(9/2) / (3/2) = 3

(9/2) * (2/3) = 3

9/3 = 3

3 = 3

The equation holds true, so our solution x = 7/2 is correct.

Key Takeaways

When solving equations with fractions, the first step is often to multiply both sides by the denominator to eliminate the fraction.
Distribute any constants to terms inside parentheses.
Isolate the variable by performing the same operations on both sides of the equation.
Always check your solution by plugging it back into the original equation.

2. Solving the Equation: x/2 + x/3 = 5/6

In this section, we’ll tackle the equation x/2 + x/3 = 5/6. This problem involves adding fractions, so we'll need to find a common denominator before we can solve for x. This is a fundamental skill in algebra, and mastering it will help you solve a wide range of problems.

Step-by-Step Solution

The first step is to find a common denominator for the fractions. The least common multiple (LCM) of 2, 3, and 6 is 6. So, we'll convert each fraction to have a denominator of 6:

(x/2) * (3/3) + (x/3) * (2/2) = 5/6

This gives us:

(3x/6) + (2x/6) = 5/6

Now that we have a common denominator, we can add the fractions:

(3x + 2x) / 6 = 5/6

Combine the terms in the numerator:

5x / 6 = 5/6

To solve for x, we can multiply both sides of the equation by 6 to eliminate the denominator:

(5x / 6) * 6 = (5/6) * 6

This simplifies to:

5x = 5

Finally, divide both sides by 5:

5x / 5 = 5 / 5

So, the solution is:

x = 1

Checking the Solution

Let’s check our solution by plugging x = 1 back into the original equation:

(1/2) + (1/3) = 5/6

To add the fractions on the left side, we need a common denominator, which is 6:

(1/2) * (3/3) + (1/3) * (2/2) = 5/6

(3/6) + (2/6) = 5/6

5/6 = 5/6

The equation holds true, so our solution x = 1 is correct.

Key Takeaways

When adding or subtracting fractions, always find a common denominator first.
The least common multiple (LCM) is often the easiest common denominator to use.
After adding or subtracting fractions, simplify the equation and solve for the variable.
Always check your solution by plugging it back into the original equation.

3. Solving the Equation: 5/(x-1) + 3/(x+1) = (7x+1)/(x^2-1)

In this section, we’ll solve the equation 5/(x-1) + 3/(x+1) = (7x+1)/(x^2-1). This problem involves adding fractions with different denominators and requires us to factor the denominator on the right side. Let's break it down step by step to make it manageable.

Step-by-Step Solution

First, notice that x^2 - 1 can be factored as (x - 1)(x + 1). This is a difference of squares, a common pattern in algebra. So, we can rewrite the equation as:

5/(x-1) + 3/(x+1) = (7x+1)/((x-1)(x+1))

Now, we need to find a common denominator for the left side of the equation. The common denominator is (x - 1)(x + 1). We'll multiply each fraction by the appropriate factor to get this common denominator:

(5/(x-1)) * ((x+1)/(x+1)) + (3/(x+1)) * ((x-1)/(x-1)) = (7x+1)/((x-1)(x+1))

This gives us:

(5(x+1))/((x-1)(x+1)) + (3(x-1))/((x-1)(x+1)) = (7x+1)/((x-1)(x+1))

Now, we can add the fractions on the left side:

(5(x+1) + 3(x-1)) / ((x-1)(x+1)) = (7x+1)/((x-1)(x+1))

Expand the numerators:

(5x + 5 + 3x - 3) / ((x-1)(x+1)) = (7x+1)/((x-1)(x+1))

Combine like terms in the numerator:

(8x + 2) / ((x-1)(x+1)) = (7x+1)/((x-1)(x+1))

Since the denominators are the same, we can set the numerators equal to each other:

8x + 2 = 7x + 1

Subtract 7x from both sides:

8x - 7x + 2 = 7x - 7x + 1

x + 2 = 1

Subtract 2 from both sides:

x + 2 - 2 = 1 - 2

So, the solution is:

x = -1

Checking for Extraneous Solutions

However, we need to be careful here! Notice that if x = -1, the denominator (x + 1) becomes zero, which makes the original equation undefined. Therefore, x = -1 is an extraneous solution, and there is actually no solution for this equation.

Key Takeaways

Factor denominators to find a common denominator when adding fractions.
Be careful of extraneous solutions when dealing with rational equations.
Always check your solutions in the original equation, especially when variables are in the denominator.

4. Solving the Inequality: (3x+1)/(x-1) ≥ 2

In this section, we’ll tackle the inequality (3x+1)/(x-1) ≥ 2. Solving inequalities is a bit different from solving equations, especially when variables are in the denominator. We need to consider the intervals where the inequality holds true, which often involves testing different regions on a number line.

Step-by-Step Solution

First, we want to get all terms on one side of the inequality. Subtract 2 from both sides:

(3x+1)/(x-1) - 2 ≥ 0

Now, we need to combine the terms on the left side into a single fraction. To do this, we need a common denominator, which is (x-1). Rewrite 2 as 2(x-1)/(x-1):

(3x+1)/(x-1) - 2(x-1)/(x-1) ≥ 0

Now we can combine the fractions:

(3x + 1 - 2(x - 1)) / (x - 1) ≥ 0

Expand the numerator:

(3x + 1 - 2x + 2) / (x - 1) ≥ 0

Combine like terms in the numerator:

(x + 3) / (x - 1) ≥ 0

Now, we need to find the critical points, which are the values of x that make the numerator or the denominator equal to zero. These are the points where the expression can change its sign:

Numerator: x + 3 = 0 => x = -3
Denominator: x - 1 = 0 => x = 1

These critical points divide the number line into three intervals: (-∞, -3), (-3, 1), and (1, ∞). We’ll test a value from each interval to see where the inequality holds true.

Interval (-∞, -3): Let’s test x = -4:

((-4) + 3) / ((-4) - 1) = (-1) / (-5) = 1/5 ≥ 0 (True)
Interval (-3, 1): Let’s test x = 0:

(0 + 3) / (0 - 1) = 3 / (-1) = -3 ≥ 0 (False)
Interval (1, ∞): Let’s test x = 2:

(2 + 3) / (2 - 1) = 5 / 1 = 5 ≥ 0 (True)

So, the inequality holds true for the intervals (-∞, -3] and (1, ∞). We include -3 because the inequality is greater than or equal to zero, but we exclude 1 because it makes the denominator zero.

Solution

The solution to the inequality is:

x ≤ -3 or x > 1

Key Takeaways

When solving inequalities with variables in the denominator, find the critical points by setting the numerator and denominator equal to zero.
Test intervals on a number line to determine where the inequality holds true.
Be careful with the endpoints of the intervals, especially when the inequality is greater than or equal to or less than or equal to.

5. Simplifying the Expression: (2x-8)/(x-2)

In this section, we're going to simplify the expression (2x-8)/(x-2). Simplifying algebraic expressions is a crucial skill, as it allows us to work with more manageable forms of equations. This often involves factoring and canceling common terms.

Step-by-Step Solution

The first step is to look for common factors in the numerator. In this case, we can factor out a 2:

(2x - 8) = 2(x - 4)

So, the expression becomes:

(2(x - 4)) / (x - 2)

Now, we look to see if there are any common factors between the numerator and the denominator that we can cancel out. In this case, there are no common factors, so the expression is already in its simplest form.

However, if the expression were (2x - 4) / (x - 2), we could factor out a 2 from the numerator:

(2(x - 2)) / (x - 2)

And then cancel the (x - 2) terms:

So, in our original expression, the simplified form is:

(2(x - 4)) / (x - 2)

Key Takeaways

Always look for common factors in the numerator and denominator.
Factor expressions to simplify them.
Cancel out common factors to reduce the expression to its simplest form.

Conclusion

Alright, guys! We’ve covered a lot in this guide, from solving basic equations to tackling inequalities and simplifying expressions. Each problem has its unique nuances, but the key is to break it down step by step and stay organized. Remember to always check your solutions and be mindful of potential pitfalls, like extraneous solutions in rational equations. Keep practicing, and you'll become a pro at solving all sorts of mathematical problems. Happy solving!