Solve 5(x-2)(x+4)>0: Inequality Solution Set

Hey math whizzes! Ever stared at an inequality like $5(x-2)(x+4)>0$ and wondered, "What's the deal with these numbers?" Don't sweat it, guys. We're about to break down how to find the solution set for this common type of problem. Inequalities can seem a bit tricky at first, but once you get the hang of them, they're super useful for understanding ranges of values where something holds true. This particular inequality involves a quadratic expression, and finding its solution set is all about understanding where the expression is positive. We'll explore the critical points, test intervals, and arrive at the correct answer.

Understanding the Inequality: $5(x-2)(x+4)>0$

Alright, let's dive deep into the nitty-gritty of our inequality: $5(x-2)(x+4)>0$. The first thing you'll notice is that we have a constant factor of 5. Since 5 is a positive number, it doesn't affect the sign of the overall expression. This means we can essentially ignore it for determining where the expression is greater than zero. Our focus will be on the factors $(x-2)$ and $(x+4)$. The core of solving this inequality lies in finding the values of $x$ that make the product of these two factors positive. Remember, a product is positive if both factors are positive OR if both factors are negative. This is a fundamental rule of multiplication that we'll be using to our advantage.

To figure out where the expression changes its sign, we need to find the roots or critical points. These are the values of $x$ where the expression equals zero. For $5(x-2)(x+4)=0$, the roots are straightforward to find. Set each factor containing $x$ to zero:

• $(x-2) = 0 ightarrow x = 2$
• $(x+4) = 0 ightarrow x = -4$

These two numbers, -4 and 2, are super important. They divide the number line into three distinct intervals: $(-\infty, -4)$, $(-4, 2)$, and $(2, \infty)$. Within each of these intervals, the sign of the expression $5(x-2)(x+4)$ will remain constant. Our job is to determine which of these intervals satisfy the condition that the expression is greater than zero (i.e., positive).

Testing the Intervals: Finding the Sweet Spot

Now that we have our critical points, -4 and 2, it's time to test the intervals they create on the number line. This is where the magic happens, guys! We need to pick a test value from each interval and plug it back into our original inequality, $5(x-2)(x+4)>0$, to see if it holds true. Remember, we only care about whether the expression is positive or negative.

Interval 1: $x < -4$ (e.g., $(-\infty, -4)$)

Let's pick a value less than -4. How about $x = -5$? Let's substitute this into our expression:

$5((-5)-2)((-5)+4) = 5(-7)(-1)$

Now, let's look at the signs: positive * negative * negative. A negative times a negative gives us a positive. So, $5(-7)(-1) = 35$. Is 35 greater than 0? Yes! This means the interval $x < -4$ is part of our solution set. We're off to a great start!

Interval 2: $-4 < x < 2$ (e.g., $(-4, 2)$)

Next up, let's grab a test value between -4 and 2. $x = 0$ is a super easy choice! Plugging it in:

$5((0)-2)((0)+4) = 5(-2)(4)$

Let's check the signs: positive * negative * positive. A negative times a positive gives us a negative. So, $5(-2)(4) = -40$. Is -40 greater than 0? No. This interval does not satisfy our inequality, so we exclude it from our solution set.

Interval 3: $x > 2$ (e.g., $(2, \infty)$)

Finally, let's pick a value greater than 2. How about $x = 3$? Substitute it in:

$5((3)-2)((3)+4) = 5(1)(7)$

Looking at the signs: positive * positive * positive. A positive times a positive is still positive. So, $5(1)(7) = 35$. Is 35 greater than 0? Yes! This means the interval $x > 2$ is also part of our solution set.

So, after testing all three intervals, we've found that the inequality $5(x-2)(x+4)>0$ is true when $x < -4$ or when $x > 2$. This gives us our solution set!

Identifying the Correct Option

Now, let's look back at the options provided to see which one matches our findings. We determined that the solution set is when $x$ is less than -4 OR when $x$ is greater than 2. Let's break down the options:

• A. {x \mid x>-4 and x<2}: This describes the interval between -4 and 2, which we found to be false. Remember, our test with $x=0$ showed this range makes the expression negative.
• B. {x \mid x<-1 or x>2}: This option has $x>2$, which matches part of our solution. However, it suggests $x<-1$. Our analysis clearly showed that the inequality holds for $x<-4$. The $-1$ here doesn't align with our critical points.
• C. {x \mid x<-2 or x>4}: Similar to option B, this has an incorrect lower bound. Our critical points were -4 and 2, not -2 and 4.
• D. {x \mid x<-4 or x>2}: This option perfectly matches our findings! It states that $x$ must be less than -4 OR greater than 2. This is exactly what our interval testing revealed.

Therefore, the correct solution set to the inequality $5(x-2)(x+4)>0$ is {x \mid x<-4 or x>2}.

Visualizing the Solution with a Number Line

Sometimes, a picture is worth a thousand words, right? Let's visualize our solution on a number line. Remember, our critical points are -4 and 2. We're looking for where the expression $5(x-2)(x+4)$ is positive.

First, mark your number line and place the critical points -4 and 2 on it. These points divide the line into three sections:

1. To the left of -4 ($x < -4$)
2. Between -4 and 2 ($-4 < x < 2$)
3. To the right of 2 ($x > 2$)

Now, recall our test results:

• For $x < -4$, the expression is positive.
• For $-4 < x < 2$, the expression is negative.
• For $x > 2$, the expression is positive.

Since our inequality is $5(x-2)(x+4)>0$ (meaning we want where it's positive), we shade the regions where the expression is positive. This means we shade the region to the left of -4 and the region to the right of 2.

Visually, this would look something like:

<--------------------(-4)--------------------(2)-------------------->

Shaded Region

<)--------------------(>

(Open circles at -4 and 2 because the inequality is strict '>' and not '>=')

This visual representation clearly shows that the solution includes all numbers less than -4 and all numbers greater than 2. This reinforces our algebraic solution and makes it much easier to grasp.

Why Other Options Are Incorrect

Let's quickly revisit why the other options just don't cut it. Understanding why they're wrong is just as important as knowing the right answer, guys!

• Option A: {x \mid x>-4 and x<2}: This represents the interval $(-4, 2)$. We tested $x=0$ and found $5(0-2)(0+4) = -40$, which is not greater than 0. So, this entire interval is incorrect.

• Option B: {x \mid x<-1 or x>2}: The part $x>2$ is correct. However, the part $x<-1$ is not entirely correct. While it includes numbers less than -4 (which are solutions), it also includes numbers between -4 and -1 (like -2 or -3). Let's test $x=-3$: $5(-3-2)(-3+4) = 5(-5)(1) = -25$, which is not greater than 0. So, $x<-1$ is too broad and incorrect.

• Option C: {x \mid x<-2 or x>4}: This option seems to have picked random numbers that aren't even the roots of our expression. Our roots are -4 and 2. The numbers -2 and 4 are irrelevant to finding the critical points for this specific inequality. Always stick to the roots you calculate!

Conclusion: Mastering Inequalities

So there you have it, folks! Solving inequalities like $5(x-2)(x+4)>0$ is all about identifying those crucial roots, testing the intervals they create, and then piecing together the parts of the number line that satisfy the inequality's condition (in this case, being positive). We found that the critical points are $x=-4$ and $x=2$, and by testing values in the intervals $(-\infty, -4)$, $(-4, 2)$, and $(2, \infty)$, we confirmed that the expression is positive when $x < -4$ or $x > 2$. This leads us directly to the correct solution set: {x \mid x<-4 or x>2}. Keep practicing these, and you'll be an inequality pro in no time! Math is all about understanding the patterns, and inequalities have some really cool ones to discover. Keep exploring, keep questioning, and most importantly, keep solving!