Solve each quadratic inequality. Use interval notation to write each solution set. 6 x^2-4 ≤5 x

Name: Problem 57, Chapter 1: Essentials of Precalculus
Uploaded: 2021-07-09T20:44:11-08:00
Duration: 3 min 39 s
Channel: Julie Silva
Description: Solve each quadratic inequality. Use interval notation to write each solution set. 6 x^2-4 ≤5 x

step 1 We're being asked to solve the given quadratic inequality. Well, first off, it has to be set to

Solve each quadratic inequality. Use interval notation to...

Key Concepts

Interval Notation

Interval notation is a standardized way of representing subsets of real numbers. Once the intervals that satisfy the quadratic inequality are determined, they are expressed in interval notation. This notation clearly communicates the solution set by indicating the endpoints and whether these endpoints are included or excluded from the interval.

Interval Testing and Sign Analysis

After identifying the zeros, the next step is to perform sign analysis on the intervals they create. This involves choosing test points from each interval to determine whether the quadratic expression is positive or negative in that region. This analysis helps determine which intervals satisfy the original inequality.

Quadratic Inequality

A quadratic inequality involves a quadratic expression and an inequality sign. The primary objective in solving such an inequality is to determine the set of x-values for which the quadratic expression is either less than, greater than, less than or equal to, or greater than or equal to zero. Understanding these inequalities requires familiarity with the properties and graphs of quadratic functions, as they are typically represented by parabolas.

Standard Form Conversion

Converting the inequality to the standard form (ax^2 + bx + c ? 0, or similar) is essential. By rearranging all terms to one side of the inequality, the problem becomes easier to analyze. This step sets the stage for finding the zeros of the quadratic equation and determining the intervals to test.

Finding Zeros of the Quadratic

Identifying the zeros, or roots, of the quadratic equation derived from setting the expression equal to zero, is crucial in solving the inequality. These zeros serve as the boundary or critical points that divide the real number line into intervals in which the quadratic expression maintains a consistent sign.

Transcript

00:01 We're being asked to solve the given quadratic inequality.

00:03 Well, first off, it has to be set to zero.

00:05 So i'm going to subtract 5x from both sides.

00:08 So in standard form, this would be 6x squared minus 5x minus 4 is less than equal to zero.

00:15 Perfect.

00:16 So now that we have it set to zero, we need to find the zeros of this quadratic, meaning we have to find when is our quadratic equal to zero.

00:24 So i'm going to do this by factoring.

00:27 Well, there is no greatest common factor.

00:28 And because the first coefficient is greater than one, i'm going to solve by a grouping method.

00:34 So i'll begin by multiplying 6 times negative 4, which is negative 24.

00:38 Then i'll find two numbers that multiply the negative 4 that will add to negative 5.

00:43 So that would be negative 8 and positive 3.

00:46 So now i'm going to rewrite my problem, but in place of negative 5x, i'm going to substitute a negative 8x plus 3x.

00:54 And then we'll bring down our first and last terms.

00:58 So now we have four terms so we can factor by grouping.

01:01 So we'll split the problem in half, and we'll factor out the greatest common factor from the first two terms, which is 2x.

01:08 When we divide both terms by 2x, that leaves us with 3x minus 4.

01:13 Then we'll factor out the greatest common factor from the second two terms, which is positive 1.

01:18 And when we divide both terms by 1, we're also left with 3x minus 4.

01:22 So notice, both factors have this greatest common factor of 3x minus 4.

01:27 So we can factor that out.

01:28 And our second factor are the terms on the outside, 2x plus 1.

01:33 Perfect.

01:34 Well, now we've factored, so now we'll set both of our factors equal to 0.

01:38 And then we'll solve both equations.

01:41 So to solve the first equation, we'll add 4 to both sides, so 3x will equal to 4, then we'll divide both sides by 3.

01:49 So x will equal to 4 thirds...

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