Solving (x+7)/(x-2) ≤ 1 A Step-by-Step Algebraic Guide

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Hey guys! Today, we're diving deep into the world of algebraic inequalities. Specifically, we're going to tackle the inequality (x+7)/(x-2) ≤ 1 algebraically. This type of problem can seem tricky at first, but with a systematic approach, you'll be solving these like a pro in no time! We'll break down each step, explain the reasoning behind it, and make sure you understand the core concepts. So, buckle up and let's get started!

Understanding the Problem

Before we jump into the solution, let's make sure we understand what the problem is asking. We have an inequality involving a rational expression, which means an expression where we have a variable in the numerator and the denominator. Our goal is to find all the values of 'x' that make the inequality true. In simpler terms, we're looking for the range of 'x' values that satisfy the condition that the fraction (x+7)/(x-2) is less than or equal to 1.

Key Concepts to Remember:

  • Inequalities: Unlike equations, inequalities use symbols like ≤ (less than or equal to), < (less than), ≥ (greater than or equal to), and > (greater than). They represent a range of values rather than a single value.
  • Rational Expressions: These are fractions where the numerator and/or the denominator are polynomials (expressions with variables and coefficients).
  • Undefined Points: Rational expressions are undefined when the denominator is equal to zero. This is a crucial point to consider when solving inequalities involving rational expressions.

Step-by-Step Solution

1. Move All Terms to the Left Side

The first step in solving this inequality is to get all the terms on one side, leaving zero on the other side. This is similar to how we solve equations, but with a slight twist for inequalities. To do this, we subtract 1 from both sides of the inequality:

(x+7)/(x-2) - 1 ≤ 0

This step is crucial because it allows us to compare the expression to zero, which is our reference point for determining the solution intervals. By having zero on one side, we can easily analyze the sign of the expression on the other side.

2. Combine the Terms into a Single Fraction

Now that we have all the terms on the left side, we need to combine them into a single fraction. To do this, we need a common denominator. In this case, the common denominator is (x-2). We rewrite 1 as (x-2)/(x-2) and then subtract:

(x+7)/(x-2) - (x-2)/(x-2) ≤ 0

Now, we can combine the numerators:

[(x+7) - (x-2)] / (x-2) ≤ 0

Simplifying the numerator, we get:

(x + 7 - x + 2) / (x-2) ≤ 0

(9) / (x-2) ≤ 0

This simplified fraction is much easier to work with. We've successfully combined the terms and now have a single rational expression.

3. Define f(x) and Identify Undefined Intervals

Let's define the left side of the inequality as f(x): f(x) = 9 / (x-2). This makes it easier to refer to the expression and analyze its behavior.

The next crucial step is to identify the intervals where f(x) is undefined. A rational expression is undefined when the denominator is equal to zero. So, we need to find the values of x that make x-2 = 0.

Setting the denominator equal to zero:

x - 2 = 0

Solving for x, we get:

x = 2

This means that f(x) is undefined when x = 2. This point is critical because it divides the number line into intervals where the sign of f(x) may change. Therefore, x = 2 is a critical value. The function is undefined at x = 2, so we must exclude this value from our solution set. This gives us the intervals (-∞, 2) and (2, ∞) to consider.

4. Determine the Sign of f(x) in Each Interval

Now, we need to determine the sign of f(x) in each of the intervals we identified. We can do this by choosing a test value within each interval and plugging it into f(x). If the result is positive, then f(x) is positive in that interval. If the result is negative, then f(x) is negative in that interval.

  • Interval (-∞, 2): Let's choose a test value, say x = 0. Plugging this into f(x):

    f(0) = 9 / (0-2) = 9 / (-2) = -4.5

    Since f(0) is negative, f(x) is negative in the interval (-∞, 2).

  • Interval (2, ∞): Let's choose a test value, say x = 3. Plugging this into f(x):

    f(3) = 9 / (3-2) = 9 / (1) = 9

    Since f(3) is positive, f(x) is positive in the interval (2, ∞).

5. Identify the Intervals That Satisfy the Inequality

Remember, we're looking for the values of x where f(x) ≤ 0. This means we want the intervals where f(x) is either negative or equal to zero. From our sign analysis, we know that f(x) is negative in the interval (-∞, 2).

Now, let's consider when f(x) = 0. For a fraction to be zero, the numerator must be zero. In our case, the numerator is 9, which is never zero. Therefore, f(x) is never equal to zero.

6. Write the Solution in Interval Notation

So, the solution to the inequality (x+7)/(x-2) ≤ 1 is the interval where f(x) is negative, which is (-∞, 2). We use a parenthesis for 2 because f(x) is undefined at x = 2, so it's not included in the solution.

Key Takeaways and Common Mistakes

  • Moving Terms: Always move all terms to one side of the inequality before proceeding.
  • Combining Fractions: Make sure to combine all terms into a single fraction to simplify the analysis.
  • Undefined Points: Don't forget to identify the points where the expression is undefined. These points are crucial for determining the solution intervals.
  • Test Values: Using test values in each interval is a reliable way to determine the sign of the expression.
  • Interval Notation: Express your final answer in interval notation, paying attention to whether the endpoints are included or excluded.

Common Mistakes to Avoid:

  • Multiplying by (x-2): A common mistake is to multiply both sides of the inequality by (x-2). However, this is dangerous because (x-2) can be positive or negative depending on the value of x. Multiplying by a negative number would require flipping the inequality sign, which is easy to forget. It's much safer to move all terms to one side and combine them into a single fraction.
  • Forgetting Undefined Points: Failing to identify the points where the expression is undefined can lead to incorrect solutions. Always check for values that make the denominator zero.
  • Incorrect Interval Notation: Make sure you use the correct notation (parentheses or brackets) to indicate whether the endpoints are included or excluded from the solution.

Practice Problems

To solidify your understanding, try solving these similar inequalities:

  1. (2x-1)/(x+3) ≥ 0
  2. (x-4)/(x+1) < 2
  3. (3x+5)/(x-2) ≤ 1

By working through these problems, you'll gain confidence in your ability to solve algebraic inequalities. Remember to follow the steps we discussed and pay attention to the key concepts.

Conclusion

Solving algebraic inequalities involving rational expressions can be a bit challenging, but by following a systematic approach, you can conquer them! The key is to move all terms to one side, combine them into a single fraction, identify the undefined points, determine the sign of the expression in each interval, and then write the solution in interval notation. Keep practicing, and you'll become a master of inequalities!

So, there you have it, guys! A comprehensive guide to solving the inequality (x+7)/(x-2) ≤ 1 algebraically. I hope this explanation has been helpful and has boosted your confidence in tackling similar problems. Remember, practice makes perfect, so keep working at it! And if you have any questions, don't hesitate to ask. Happy solving!