Solve each inequality. Write the solution set in interval notation. (3 x-12)(x+5)(2 x-3) ≥0

Name: Problem 32, Chapter 11: Beginning and Intermediate Algebra
Uploaded: 2020-03-31T21:11:57-08:00
Duration: 7 min 10 s
Channel: Deandre Johnson
Description: Solve each inequality. Write the solution set in interval notation. (3 x-12)(x+5)(2 x-3) ≥0

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

Key Concepts

Polynomial Inequalities

A polynomial inequality involves finding the set of values of the variable that make a polynomial expression greater than, less than, or equal to zero. This concept includes understanding how the terms and degrees of the polynomial affect its behavior over the real number line.

Critical Points

Critical points are values of the variable where any factor of the expression equals zero. These points are significant because they are potential boundaries between intervals where the polynomial changes sign, and they are used to divide the number line into segments for analysis.

Test Intervals

Once the critical points are identified, the number line is partitioned into intervals. A representative value (test point) from each interval is used to determine the sign of the polynomial expression within that interval, which helps in constructing the solution set for the inequality.

Sign Chart

A sign chart is a visual tool that organizes the sign (+ or -) of each factor and the overall expression across different intervals. It simplifies determining where the polynomial is positive, negative, or zero, which is essential for solving inequalities.

Interval Notation

Interval notation is the standard form for expressing sets of numbers. It uses brackets to denote inclusion of endpoints and parentheses for exclusion, allowing a concise and precise description of the solution set for inequalities.

Transcript

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

00:07 So essentially what we want to find is when this function is greater than are equal to zero.

00:13 To begin, we're just going to rewrite this function equal to zero so that we can solve for x.

00:22 And then how we solve for x, of course, is just set each of these individual functions equal to zero.

00:30 3x minus 12 equals 0, x plus 5 equals 0, and 2x minus 3 equals 0.

00:38 And then we can just solve for x like so.

00:41 3x equals 12 by adding 12 to both sides, divide by 3, and x is equal to 4.

00:47 Right here, we can just subtract 5 from both sides, x equals negative 5, and then finally add 3 to both sides for this one, and divide by 2 on both sides, and we see that x is equal to 3 over 2.

01:00 Cool.

01:01 So these are our solutions to our function.

01:04 And basically, if we plug these in, that'll make the function equal to zero.

01:09 But we want to, we need to find when the function is also greater than zero.

01:14 So to do that, we're going to open up a new page and rewrite our function here at the top, minus 12 times x plus 5 times 2x.

01:33 It's greater than or equal to now, we need to check the entire number set to see where the function is greater than zero.

01:41 To do that, we're going to put all of our solutions on the number line, from least to greatest.

01:46 So, negative 5, positive 3 over 2, and positive 4.

01:55 So the intervals we're going to check are from negative infinity all the way up to negative 5, negative 5, up to 3 over 2, 3 over 2 to 4.

02:09 And 4 all the way up to positive infinity.

02:13 Now, in order to check these intervals, what we're going to do is choose a number that's included inside of that interval and plug it into this function up here.

02:22 So for our first interval, we'll check the number negative 6.

02:27 And we'll just do that right here.

02:31 So 3 times negative 6 minus 12 times negative 6 plus 5 times.

02:40 2 times negative 6 minus 3.

02:45 And that will get us negative 18 minus 12, or negative 30, times negative 6 plus 5, or negative 1, and negative 12 minus 3, or negative 15.

03:06 And so we have a negative times a negative, which will give us a positive, and then a positive times a negative, which will get us a negative...

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