Inequalities Involving Radicals

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Do curso por University of California, Irvine

Pre-Calculus: Functions

298 classificações

University of California, Irvine

Pre-Calculus: Functions

298 classificações

This course covers mathematical topics in college algebra, with an emphasis on functions. The course is designed to help prepare students to enroll for a first semester course in single variable calculus. Upon completing this course, you will be able to: 1. Solve linear and quadratic equations 2. Solve some classes of rational and radical equations 3. Graph polynomial, rational, piece-wise, exponential and logarithmic functions 4. Find integer roots of polynomial equations 5. Solve exponential and logarithm equations 6. Understand the inverse relations between exponential and logarithm equations 7. Compute values of exponential and logarithm expressions using basic properties

Na lição

Linear Equations and Inequalities. Absolute Value Equations and Inequalities.

In this module, we will learn to solve linear and absolute value equations and inequalities. We will also explore a range of story problems which can be solved utilizing linear equations.

Absolute Value Equations 6:35

Absolute Value Inequalities 6:32

Inequalities Involving Radicals 5:49

Conheça os instrutores

Dr. Sarah Eichhorn
Associate Dean, Distance Learning & Tenured Lecturer
Dr. Rachel Cohen Lehman

[SOUND]. Let's solve Inequalities that Involve

Radicals. [SOUND].

For example let's solve the following inequality for x.

Alright, so what is the left hand side here equal.

Most students will say that it equals x - 6, which is not true.

In fact, it equals the absolute value of x - 6.

That is, the square root of something ^2 is only equal to that thing.

If that thing, is greater than or equal to 0 , otherwise, it's the negative of

it. Or, the square root of y^2 is equal to

the absolute value of y. For example.

The square root say of 2 ^ 2 is equal to the square root of 4.

Which is 2. So yes, the square root of y^2 is y.

But what about the square root say of -2 ^ 2.

If that y is negative, this is the square root of 4, which is 2.

So these aren't equal, they're the negatives of each other.

So be very careful here, it's a very common mistake.

Therefore the inequality, that we need to solve, is at the absolute value of |x -

6| <= 9. And remember when solving absolute value

inequalities, this means that negative 9 has to be less than or equal to x - 6 has

to be less than or equal to 9. So this translates into this compound

inequality. And now we can add 6 everywhere.

Which gives us, -9 + 6 is less than or = to x.

Is less than or = to 9 + 6. Or -3 is less than or = to x, is less

than or = to 15. Which we can write in integral notation

as the integral negative 3 up to 15. Or we could even graph it, the solution

set would be the integral from -3 including -3 all the way up to 15.

Including 15. So just be really careful that you use

the absolute value here. Alright.

[SOUND] Let's see another example. Let's solve this inequality for y.

Again we have the square root of something squared.

And be careful the number is the absolute value of that thing.

Which needs to be greater than 5. So when solving absolute value

inequalities, if we have the absolute value of something as larger than 5, then

that means that, that quantity is either larger than 5 or that quantity.

Is smaller than -5. And now we'll solve this compound

inequality, by working each side separately.

Solving this first inequality, we subtract 3 from both sides, which gives

us -2y Is greater than 5 - 3 or negative 2y is greater than 2.

And now we'll divide both sides by negative 2.

But remember we have to flip or reverse the direction of this inequality.

Which gives us y is less than negative 1. Alright, and what about the second

inequality? Again, we'll subtract 3 from both sides, which gives us negative 2y is

less than negative 5 - 3. Or -2y is less than -8.

Again, dividing by -2 and flipping or reversing the direction of our inequality

gives us that y is greater than 4. And because these inequalities are joined

by the word or, our solution here will be the set of all Y that make at least 1 of

these inequalities true. So graphing this gives us the following.

Let's say this is negative 1. Less than -1, we do not want to include

-1, so we put an open circle and less than, and we go to the left.

And let's say is this 4. We do not want to include 4, so we put an

open circle greater than and we go to the right, which will be our answer here.

And in an interval notation. We could write negative infinity up to

-1, open parentheses, because we don't want to include -1.

Union, again open parenthese at 4 up to infinity.

So just be very careful that you use this absolute value here when starting Thank

you, and we'll see you next time. [MUSIC]

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