Solve each inequality by using the method of your choice. State the solution set in interval not

Solve each inequality by using the method of your choice....

Key Concepts

Quadratic Inequalities

Quadratic inequalities involve expressions set as greater than or less than zero (or another value) and require determining for which values of the variable the quadratic expression satisfies the inequality. Typically, this is achieved by finding the roots of the corresponding quadratic equation and then analyzing the sign of the quadratic expression in the intervals determined by these roots.

Finding Roots Using the Quadratic Formula

The quadratic formula is a universal method for solving quadratic equations. When a quadratic expression does not factor easily, the quadratic formula, x = (-b ± ?(b² - 4ac)) / (2a), provides a straightforward way to find the roots, which are crucial for dividing the number line into intervals for the inequality analysis.

Sign Analysis

Once the roots of the quadratic equation are found, the number line can be divided into segments where the sign of the quadratic expression (positive or negative) is constant. Testing a value from each interval helps determine which intervals satisfy the original inequality.

Interval Notation

Interval notation is a concise method used to describe the set of all numbers that satisfy a particular condition. This mathematical notation uses endpoints combined with brackets or parentheses to indicate whether endpoints are included or excluded in the solution set.

Graphical Representation on a Number Line

Graphical representation involves plotting the solution set of an inequality on a number line. This visual tool uses open or closed circles at critical points (roots) based on whether the endpoints are included, and it shades the regions of the number line that meet the inequality conditions.

Transcript

00:01 We want to solve this inequality using any method of our choice and then state the solution in interval notation as well as graph it.

00:10 Well, i think this will factor.

00:12 So let's go ahead and do that first.

00:14 So we can use the pq method.

00:18 So we multiply these together to get negative 12.

00:20 And then we ask ourselves what factors of negative 12 will add up to negative 4.

00:27 Well that looks like it would be negative 6 and positive 2 because when we add those we get negative 4 so we would want to break that up and remember we're doing this because we want to figure out what the zeros of our quadratic is so it would be 3x squared minus 6x plus 2 x minus 4 and out of this first term we can factor out 3x so we get 3x x minus 2 and then out of our second term we can just factor out of 2 so it would be plus 2 x minus 2 and then we can factor out x minus 2 from each of these terms so it's going to be x minus 2 3x plus 2 and so this should be less than or equal to 0 and the values that are going to make for quadratic equal to zero or x is equal to two and x is equal to negative two thirds it looks like if we use the zero product property now let's go ahead and bring up our graph so we have negative two thirds here two over here now to test we want to go ahead and test a number smaller than negative two thirds first let's just plug in negative one now remember we're going to plug it into our quadratic up here.

02:10 So when x is equal to negative 1, we'll get 3 times negative 1 squared minus 4 times negative 1 minus 4, which would give us 3 plus 4 minus 4.

02:25 So the 4 and negative 4 counts out and we're just left for 3.

02:28 But recall, the only thing we really care about is that it's positive on that interval.

02:32 Then we test the number between negative two -thirds and two.

02:36 So let's do zero.

02:40 And when x is equal to zero, we know it should just give us c here...

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