Solve each inequality. Write the solution set in interval notation. ((2 x-3)^2)/(x)<0

Name: Problem 54, Chapter 11: Beginning and Intermediate Algebra
Uploaded: 2020-04-01T00:42:59-08:00
Duration: 4 min 42 s
Channel: Deandre Johnson
Description: Solve each inequality. Write the solution set in interval notation. ((2 x-3)^2)/(x)<0

Solve each inequality. Write the solution set in interval...

Key Concepts

Rational Inequalities

Rational inequalities involve expressions that are ratios of polynomials. Solving these inequalities requires determining where the expression is positive, negative, or zero, taking into account the behavior of both the numerator and the denominator. This approach is vital because the sign of the rational expression can change only at certain key values, which then define the intervals to be tested.

Critical Points

Critical points are the values where the numerator is zero (making the entire expression zero) or where the denominator is zero (where the expression is undefined). These points are essential because they mark the potential boundaries of solution intervals and indicate where the expression might change its sign. Understanding and finding these points is crucial in analyzing and solving rational inequalities.

Sign Analysis

Sign analysis is the process of determining the sign (positive or negative) of a rational expression in intervals between the critical points. By selecting test values in each interval, one can evaluate the expression's sign and determine where it satisfies the inequality. This step ensures that each interval is correctly classified relative to the inequality being solved.

Interval Notation

Interval notation is a concise way to represent the set of all values that satisfy an inequality. It uses parentheses and brackets to indicate open or closed endpoints, respectively. Mastering interval notation is important for clearly communicating the solution set of an inequality after performing the necessary algebraic and sign analysis steps.

Transcript

00:01 All right, for this problem, we're asked to solve the inequality and write the solution in interval notation.

00:06 Our first step is going to be to solve for the critical numbers.

00:10 And how we do that is we're going to set the numerator and the denominator equal to zero and solve for x.

00:16 So we'll start with the numerator.

00:18 2x minus 3 squared equals 0.

00:21 We'll take the square root on both sides, and we can see that 2x minus 3 is equal to 0.

00:27 We just add three to both sides like so, and then divide by two.

00:33 And we can see that x is equal to 3 over 2.

00:35 This is our first critical number.

00:38 To find our second critical number, we're going to set the denominator equal to 0.

00:43 X equals 0.

00:46 Cool.

00:47 That was easy.

00:48 The denominator is also special in that it tells us what we can't have x equal.

00:53 So if we try to plug in zero into this function here, we'll see that the denominator will be zero.

01:03 And that's something that we don't want because if the denominator is equal to zero, then the entire expression is going to be undefined.

01:11 And that is no bueno.

01:12 So we're going to state that x cannot equal zero.

01:18 Nonetheless, zero is still one of our critical numbers.

01:21 So we're going to take it and 3 over 2 and put them on a number line.

01:28 0 and 3 over 2.

01:31 What this number line is going to help us do is it's going to help us figure out on which intervals.

01:36 The function is less than 0.

01:41 So to figure out when the function is less than 0, we're going to check specific intervals.

01:46 So our first interval is going to be from negative infinity to 0.

01:50 Our second interval is going to be from 0 to 3 over 2, and our final interval is going to be from 3 over 2 to positive infinity.

01:58 In order to check these intervals, we're just simply going to choose a number that's included in the interval and plug it into our equation here, 2x minus 3 squared over x...

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