Solving Inequalities In R: A Step-by-Step Guide

Hey guys! Today, we're diving into the world of inequalities and how to solve them using the power of R. We'll break down each problem step-by-step, making sure even if you're new to this, you'll get the hang of it. Let's get started! Inequalities are mathematical statements that compare two expressions using symbols like '<' (less than), '>' (greater than), '≤' (less than or equal to), and '≥' (greater than or equal to). Solving an inequality means finding all the values of the variable (usually 'x') that make the inequality true. The main difference between solving equations and inequalities is that when you multiply or divide both sides of an inequality by a negative number, you have to flip the inequality sign. Keep this in mind as we tackle each problem. So, grab your notebooks and let's solve some inequalities in R!

a) Solving 2√10x < -8√15

Alright, let's kick things off with the first inequality: 2√10x < -8√15. Our goal here is to isolate 'x'. Here's how we do it, piece by piece. First, let's get rid of the coefficient of 'x', which is 2√10. To do this, we'll divide both sides of the inequality by 2√10. Now, let's actually perform the division. Dividing both sides by 2√10 gives us:

x < (-8√15) / (2√10)

Now, let's simplify the right-hand side. We can simplify the fraction by dividing both the numerator and the denominator by 2:

x < (-4√15) / √10

Next, let's rationalize the denominator by multiplying the numerator and denominator by √10:

x < (-4√15 * √10) / (√10 * √10)

Which simplifies to:

x < (-4√150) / 10

We can further simplify the square root of 150 because it contains a perfect square: √150 = √(25 * 6) = 5√6. Let’s substitute it back:

x < (-4 * 5√6) / 10

x < (-20√6) / 10

Finally:

x < -2√6

So, the solution to the inequality is x < -2√6. This means any value of 'x' that is less than -2√6 will satisfy the original inequality. To get a numerical approximation for the solution, we can calculate the value of -2√6, which is approximately -4.898. Therefore, x must be less than -4.898.

b) Solution steps unavailable

This problem's solution steps were not provided in the original request. We will proceed to the next available problem.

c) Solving -4√7x > -8√21

Let's solve -4√7x > -8√21. To isolate 'x', we'll divide both sides by -4√7. Remember, when we divide or multiply an inequality by a negative number, we flip the inequality sign! Here's what that looks like: Dividing both sides by -4√7 and flipping the sign we get:

x < (-8√21) / (-4√7)

Next, let's simplify the right side of the inequality. First, divide both numerator and denominator by -4:

x < (2√21) / √7

Now, we can simplify the fraction further by dividing √21 by √7:

x < 2√(21/7)

x < 2√3

So, the solution to this inequality is x < 2√3. This means any value of 'x' that is less than 2√3 will satisfy the original inequality. The approximate value of 2√3 is 3.464. Therefore x must be less than 3.464.

d) Solving -3√12x ≥ 12√15

Now, let's tackle -3√12x ≥ 12√15. To isolate 'x', we'll divide both sides by -3√12. Again, remember that when we divide by a negative number, we flip the inequality sign! Here’s how it goes: dividing both sides by -3√12 gives us:

x ≤ (12√15) / (-3√12)

Next, let's simplify the right-hand side. Divide 12 by -3:

x ≤ (-4√15) / √12

Now, let's simplify the square root of 12, since √12 = √(4 * 3) = 2√3. Substituting that into the equation:

x ≤ (-4√15) / (2√3)

Simplify by dividing -4 and 2:

x ≤ (-2√15) / √3

Now, let's rationalize the denominator by multiplying the numerator and denominator by √3:

x ≤ (-2√15 * √3) / (√3 * √3)

Which simplifies to:

x ≤ (-2√45) / 3

We can simplify the square root of 45: √45 = √(9 * 5) = 3√5. Let's substitute that in:

x ≤ (-2 * 3√5) / 3

x ≤ (-6√5) / 3

Finally:

x ≤ -2√5

So, the solution to the inequality is x ≤ -2√5. This means any value of 'x' that is less than or equal to -2√5 will satisfy the original inequality. The approximate value of -2√5 is -4.472. Thus, x must be less than or equal to -4.472.

e) Solving -12√40 < 24√2x

Alright, let’s work through -12√40 < 24√2x. To isolate 'x', we need to divide both sides by 24√2. The inequality sign remains the same in this case, as we're dividing by a positive number. Here’s the breakdown: dividing both sides by 24√2:

(-12√40) / (24√2) < x

Now, let's simplify the left side of the inequality. First, simplify the fraction by dividing -12 by 24:

(-√40) / (2√2) < x

We can simplify the square root of 40: √40 = √(4 * 10) = 2√10. Let's substitute that in:

(-2√10) / (2√2) < x

Simplify by dividing 2 by 2:

-√10 / √2 < x

Let’s rationalize the denominator:

(-√10 * √2) / (√2 * √2) < x

This simplifies to:

-√20 / 2 < x

We can simplify the square root of 20: √20 = √(4 * 5) = 2√5. Substituting it back in:

(-2√5) / 2 < x

Finally:

-√5 < x

So, the solution to this inequality is x > -√5. This means any value of 'x' that is greater than -√5 will satisfy the original inequality. The approximate value of -√5 is -2.236. So, x must be greater than -2.236.

f) Solving -2√45 > -3√10x

Last but not least, let's solve -2√45 > -3√10x. To isolate 'x', we'll divide both sides by -3√10. Remember, since we are dividing by a negative number, we have to flip the inequality sign! Here’s how it goes:

x < (-2√45) / (-3√10)

Let's simplify the right-hand side. The negative signs cancel each other out:

x < (2√45) / (3√10)

We can simplify the square root of 45: √45 = √(9 * 5) = 3√5. Substituting:

x < (2 * 3√5) / (3√10)

x < (6√5) / (3√10)

Divide 6 by 3:

x < (2√5) / √10

Now, let’s rationalize the denominator by multiplying the numerator and denominator by √10:

x < (2√5 * √10) / (√10 * √10)

This simplifies to:

x < (2√50) / 10

Simplify the square root of 50: √50 = √(25 * 2) = 5√2. Substitute it in:

x < (2 * 5√2) / 10

x < (10√2) / 10

Finally:

x < √2

So, the solution to this inequality is x < √2. This means any value of 'x' that is less than √2 will satisfy the original inequality. The approximate value of √2 is 1.414. Therefore, x must be less than 1.414.

And that's it, guys! We have successfully solved all the inequalities. Remember to always pay attention to the inequality sign and flip it when multiplying or dividing by a negative number. Keep practicing, and you'll become a pro in no time. If you want, solve the same problems using a calculator and check each result. Happy solving!