Solve each quadratic inequality. Graph the solution set and write the solution in interval notat

Solve each quadratic inequality. Graph the solution set and...

Key Concepts

Interval Notation

Interval notation is a compact way to represent sets of numbers on the real number line that satisfy an inequality. It clearly communicates the boundaries of the solution set, using parentheses for strict inequalities and brackets for inclusive inequalities.

Sign Analysis (Test Intervals)

Once the zeros are identified, the number line is divided into intervals. Testing a point from each interval helps determine whether the quadratic expression is positive or negative in that interval, thus identifying the solution set of the inequality.

Graphical Representation

Graphing involves illustrating the solution set on a number line, where open or closed circles denote excluded or included endpoints. This visual aid helps in understanding and verifying the range of values that satisfy the inequality.

Standard Form and Rearrangement

Writing the inequality in standard form, where one side is zero, is a fundamental step. This rearrangement allows you to identify the structure of the quadratic expression and prepare it for methods such as factoring or using the quadratic formula.

Quadratic Inequality

A quadratic inequality involves a quadratic expression set in relation to zero (or another value) using inequalities like <, >, ?, or ?. Solving such inequalities typically requires expressing the inequality in standard form and determining where the quadratic function is positive or negative.

Finding Zeros of the Quadratic

The roots or zeros of the quadratic, found by factoring or applying the quadratic formula, split the real number line into intervals. The sign of the quadratic expression can change at these points, making them crucial for determining where the inequality holds.

Transcript

00:01 Alright, we want to solve this quadratic inequality, graph the solution set, and write it in an interval notation.

00:09 So first thing we want to do is move the negative 16 over, so we add 16 to both sides.

00:18 And we need to find what our solution are, so we know that 8 and 2, when you multiply them, they equal 16, and then when you add them, they equal 10, so that works in the inequality so then we know that our solutions are then negative two and negative eight when we graph this since it's an inequality that does not have an equal sign to it they're going to be open circles so there's a negative two and then sorry the negative eight's right there we need to figure out is the inequality going to go within, so is it going to go on the inside of the negative 2 and the negative 8, or is it going to be on the outside of the negative 2 and the negative 8, so that the graph is going to be less than 0.

01:24 So it's going to be on the negative y axis.

01:26 It's going to be on this negative y axis.

01:29 So i just pick a number in between these two to figure out is it going to be in between.

01:36 Between those two solutions.

01:39 So i'm gonna pick negative three, because it's in between negative two and negative eight.

01:43 And i'll just plug that into this equation here.

01:48 So negative three squared plus 10 times negative three, is that less than 16? so negative three squared is nine minus 30, which gives you negative 21, which is indeed less than negative 16.

02:11 So we then know that it's going to actually be in between the negative 2 and negative 8...

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