Integer Solutions For 2x + 6/(14 - 2x) > 0 A Mathematical Exploration

by ADMIN 70 views

Hey there, math enthusiasts! Ever stumbled upon an inequality that looks like a mathematical puzzle? Well, today we're diving deep into one such intriguing problem: finding the number of integer solutions for the inequality 2x + 6/(14 - 2x) > 0. Buckle up, because we're about to embark on a journey filled with algebraic manipulations, critical thinking, and a dash of mathematical finesse. This exploration isn't just about crunching numbers; it's about understanding the underlying concepts and problem-solving strategies that make math so captivating.

Unraveling the Inequality

Let's start by dissecting the inequality 2x + 6/(14 - 2x) > 0. At first glance, it might seem a bit intimidating with its combination of linear terms and fractions. But don't worry, guys, we're going to break it down step by step. Our primary goal here is to isolate 'x' and determine the range of values that satisfy the inequality. However, we need to be cautious about the denominator (14 - 2x), as it cannot be equal to zero. This is a crucial point to remember as it will influence our solution set. We'll need to consider this restriction while manipulating the inequality to avoid any mathematical pitfalls. So, let's roll up our sleeves and get started with the algebraic gymnastics!

The Art of Algebraic Manipulation

The first step in solving this inequality is to get rid of the fraction. To do this, we'll multiply both sides of the inequality by (14 - 2x). But hold on! Remember that the sign of (14 - 2x) matters. If (14 - 2x) is positive, the inequality sign remains the same. If it's negative, we need to flip the inequality sign. This is a critical step, guys, and we can't afford to overlook it. So, we'll consider two cases:

  • Case 1: (14 - 2x) > 0

    If (14 - 2x) is positive, multiplying both sides by it preserves the inequality. This gives us:

    (2x + 6)(14 - 2x) > 0

    Now, let's simplify this expression. Expanding the product, we get:

    28x - 4x² + 84 - 12x > 0

    Rearranging the terms, we have a quadratic inequality:

    -4x² + 16x + 84 > 0

    To make things easier, we can divide both sides by -4 (and remember to flip the inequality sign since we're dividing by a negative number):

    x² - 4x - 21 < 0

    Now, we need to factor this quadratic expression. Factoring gives us:

    (x - 7)(x + 3) < 0

    To find the solution to this inequality, we need to determine the intervals where the product (x - 7)(x + 3) is negative. This happens when the two factors have opposite signs. So, we analyze the sign of each factor:

    • x - 7 < 0 when x < 7
    • x + 3 > 0 when x > -3

    Thus, the solution for this case is -3 < x < 7. But remember, this is under the condition that (14 - 2x) > 0. Let's solve this condition:

    14 - 2x > 0

    14 > 2x

    x < 7

    So, the solution for this case is the intersection of -3 < x < 7 and x < 7, which is -3 < x < 7.

  • Case 2: (14 - 2x) < 0

    If (14 - 2x) is negative, multiplying both sides by it reverses the inequality sign. This gives us:

    (2x + 6)(14 - 2x) < 0

    We already expanded this product in Case 1, so we have:

    -4x² + 16x + 84 < 0

    Dividing both sides by -4 (and flipping the inequality sign again), we get:

    x² - 4x - 21 > 0

    Factoring, we have:

    (x - 7)(x + 3) > 0

    This time, we need to find the intervals where the product (x - 7)(x + 3) is positive. This happens when both factors have the same sign:

    • Both factors are positive: x - 7 > 0 and x + 3 > 0, which means x > 7
    • Both factors are negative: x - 7 < 0 and x + 3 < 0, which means x < -3

    So, the solution for this part is x < -3 or x > 7. Now, let's consider the condition (14 - 2x) < 0:

    14 - 2x < 0

    14 < 2x

    x > 7

    The solution for this case is the intersection of (x < -3 or x > 7) and x > 7, which is x > 7.

Combining the Solutions

Now, we need to combine the solutions from both cases. In Case 1, we found that -3 < x < 7. In Case 2, we found that x > 7. So, the complete solution to the inequality is -3 < x < 7 or x > 7. However, we must remember our initial restriction: 14 - 2x ≠ 0, which means x ≠ 7. Therefore, the final solution is -3 < x < 7 or x > 7. This seems counterintuitive, but remember we are looking for integer solutions. So we need to explicitly look at integers greater than 7.

Counting the Integer Solutions

We've successfully navigated the algebraic maze and arrived at the solution: -3 < x < 7 or x > 7. But the question asks for the number of integer solutions. So, our next task is to count the integers that fall within these ranges. Let's break it down:

Integer Solutions in the Interval -3 < x < 7

This interval includes all integers strictly greater than -3 and strictly less than 7. So, the integers in this range are: -2, -1, 0, 1, 2, 3, 4, 5, and 6. Counting them, we find there are 9 integer solutions in this interval. Make sure you guys are paying attention to the strict inequalities; it's a common mistake to include the endpoints when they shouldn't be.

Integer Solutions for x > 7

This condition includes all integers greater than 7. So, the integers in this range are: 8, 9, 10, 11, and so on. Notice that this range is infinite – there are infinitely many integers greater than 7. This is a crucial observation, guys, as it directly impacts the total number of integer solutions.

The Grand Finale: Total Number of Integer Solutions

Now, let's put it all together. We have 9 integer solutions in the interval -3 < x < 7, and infinitely many integer solutions for x > 7. Therefore, the total number of integer solutions for the inequality 2x + 6/(14 - 2x) > 0 is infinite. This might seem a bit surprising, especially after all the algebraic manipulations we've done. But it highlights the importance of carefully considering all parts of the solution and not overlooking infinite possibilities.

Wrapping Up

So, there you have it, guys! We've successfully navigated the inequality 2x + 6/(14 - 2x) > 0 and determined that it has infinitely many integer solutions. This problem was a fantastic exercise in algebraic manipulation, critical thinking, and careful consideration of different cases. Remember, guys, math isn't just about finding the right answer; it's about the journey of problem-solving and the insights we gain along the way. Keep exploring, keep questioning, and keep having fun with math!

I hope this explanation was clear and helpful. If you have any more math puzzles you'd like to unravel, bring them on! Let's continue our mathematical adventures together.