Solve the inequality. Then graph the solution set. (x+12)/(x+2) ≥3

Name: Problem 44, Chapter 2: Precalculus
Uploaded: 2021-05-27T19:09:27-08:00
Duration: 7 min 37 s
Channel: Guadalupe Oldham
Description: Solve the inequality. Then graph the solution set. (x+12)/(x+2) ≥3

Solve the inequality. Then graph the solution set....

Key Concepts

Rational Inequality

A rational inequality is an inequality that involves a rational expression, which is the ratio of two polynomials. Solving such inequalities requires careful handling because the expression can change sign based on the values of the variable, and division by zero must be avoided.

Domain Restrictions

When working with rational expressions, it is crucial to identify values that make the denominator zero because these values are not included in the domain of the function. Establishing domain restrictions prevents invalid operations and helps in correctly determining the solution set of the inequality.

Critical Points

Critical points are the values of the variable that make either the numerator or the denominator equal to zero. These points are significant because they are potential boundaries where the sign of the rational expression can change. They also aid in segmenting the number line into intervals for further analysis.

Common Denominator and Inequality Transformation

Transforming a rational inequality often involves rewriting the inequality so that all terms are on one side and obtaining a common denominator. This technique helps in simplifying the inequality to a single rational expression whose sign can then be analyzed systematically.

Sign Chart Technique

The sign chart technique is used to determine the sign of a rational expression over different intervals of the number line. By testing sample points from each interval created by the critical points, one can decide where the inequality holds true, which helps in graphing the solution set.

Transcript

00:01 To solve a polynomial inequality or a rational inequality, our first step is to place our inequalities in general form, which means that we'll have our function here on one side and then our sign here, which can be less than equal to, greater than, and doesn't matter on there whatever that equation gives you, that inequality gives you.

00:51 And on the other side, we'll have it set to zero.

00:53 So if this side is not equal to zero, we'll subtract this number from both sides.

01:04 And then we'll have our general form.

01:07 Our next step would be to find the zeros of our function or where it's undefined.

01:18 Zeros and undefined values.

01:29 So to do this, we need a factor.

01:31 Sometimes we need to use our function x equals negative b plus or minus the square root of 4ac.

01:45 Sorry.

01:46 A here there.

01:47 I skip the b squared minus 4ac all over 2a.

01:54 And this will help us find those zeros as well and some later on problems.

02:02 Here then once we have our zeros we call them our key numbers and after we have labeled our key numbers we will use the key numbers to create our test intervals and finally we will test the intervals with chosen x values within the interval.

03:03 To solve the rational inequality, x plus 12 over x plus 2 is greater than or equal to 3, we'll first put this in our general form and subtract the 3 on both sides, which will give us x plus 12 over x plus 2 minus 3 which is greater than or equal to 0.

03:35 Now we'll want to find our least common denominator...

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