Write the solution set of each inequality if x is an element of the set of integers. x^2-x<6

Name: Problem 12, Chapter 1: Algebra 2 and Trigonometry
Uploaded: 2019-07-31T22:12:41-08:00
Duration: 4 min 53 s
Channel: M S
Description: Write the solution set of each inequality if x is an element of the set of integers. x^2-x<6

step 1 So we have x squared minus x is less than 6. So because we know we want to factor a trinomial, w

Write the solution set of each inequality if x is an element...

Key Concepts

Factoring Quadratic Expressions

Factoring is the process of rewriting a quadratic expression as a product of two simpler expressions. This technique makes it easier to find the roots of the quadratic, which are the critical points at which the expression equals zero.

Domain Restrictions to Integers

When a problem specifies that the variable is an integer, the continuous range of solutions must be adjusted to include only whole numbers. This involves identifying all integer values within the interval that satisfies the inequality.

Quadratic Inequalities

Quadratic inequalities involve a quadratic expression set in relation to zero using inequality signs. The solution is typically found by determining the roots of the quadratic equation and then analyzing the intervals defined by these roots to identify where the inequality holds.

Critical Points and Interval Testing

Critical points, obtained from setting the factors to zero, divide the number line into intervals. By testing values in these intervals, one can determine which regions satisfy the inequality.

Transcript

00:01 So we have x squared minus x is less than 6.

00:05 So because we know we want to factor a trinomial, we need to move this 6 to the other side, so i'm going to subtract 6 from both sides.

00:11 So we'll have x squared minus x minus 6 is less than 0.

00:16 So to factor this, we need to split apart our middle term here.

00:21 So we need to multiply our coefficient times our last term, and that'll give us negative 6.

00:25 And we need them to add to negative 1.

00:29 So we're going to list the factors of 6.

00:31 And 6 and 2 and 3, and we know that we need one of the numbers in each pair to be negative.

00:36 And it also needs to add to negative 1.

00:38 So after testing a bunch of combinations, we'll get positive 2 and negative 3 as our pair.

00:48 So now we just need to split apart our middle term.

00:51 So we'll have x squared minus 2x, no plus 2x, minus 3x, and then minus 6 less than 0.

01:06 So we're going to group the first two, then group the second two.

01:10 So we're going to factor out the greatest common factor.

01:12 And from the first one, we can factor on an x, so we'll have x plus 2.

01:15 And from the second one, we can factor out a negative 3, so we'll have x plus 2.

01:20 It's greater than 0.

01:22 And it's less than 0, sorry.

01:25 So because this is the same as this, we know that x plus 2 times x minus 3 is less than 0.

01:32 So solving inequalities takes a further step than just when the trinomule is equal to zero.

01:42 Because we're looking for a range of values, so we need to find a range that meets all the qualifications.

01:51 So because we know that this product is basically negative, less than zero means it's negative.

01:56 The only way to get a negative product is if this one is positive and this one is negative.

02:02 That would give us a negative.

02:04 Or this one is negative and this one is positive and that will also give us a negative.

02:10 So i'm going to say x plus 2 is positive and x minus 3 is negative.

02:20 And we also need to say x plus 2 is negative and x minus 3 is positive.

02:31 So when i say like for instance this one here, greater than 0 just means it's positive.

02:37 So i just followed the signs...

Please give Ace some feedback