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## Graphing Quadratic Inequalities

A quadratic inequality of the form

y

>

a

x

2

+

b

x

+

c

(or substitute

<

,

≥

or

≤

for

>

) represents a region of the plane bounded by a

parabola

.

To graph a quadratic inequality, start by graphing the parabola. Then fill in the region either above or below it, depending on the inequality.

If the inequality symbol is

≤

or

≥

, then the region includes the parabola, so it should be graphed with a solid line.

Otherwise, if the inequality symbol is

<

or

>

, the parabola should be drawn with a dotted line to indicate that the region does not include its boundary.

Example:

Graph the quadratic inequality.

y

≤

x

2

−

x

−

12

The related equation is:

y

=

x

2

−

x

−

12

First we notice that

a

, the coefficient of the

x

2

term, is equal to

1

. Since

a

is positive, the parabola points upward.

The right side can be factored as:

y

=

(

x

+

3

)

(

x

−

4

)

So the parabola has

x

-intercepts

at

−

3

and

4

. The

vertex

must lie midway between these, so the

x

-coordinate of the vertex is

0.5

.

Plugging in this

x

-value, we get:

y

=

(

0.5

+

3

)

(

0.5

−

4

)

y

=

(

3.5

)

(

−

3.5

)

y

=

−

12.25

So, the vertex is at

(

0.5

,

−

12.25

)

.

We now have enough information to graph the parabola. Remember to graph it with a solid line, since the inequality is “less than or equal to”.

Should you shade the region inside or outside the parabola? The best way to tell is to plug in a sample point.

(

0

,

0

)

is usually easiest:

0

≤

?

0

2

−

0

−

12

0

≤

−

12

So, shade the region which does

not

include the point

(

0

,

0

)

.

Quadratic inequalities (video) | Khan Academy – let's say that we want to solve the inequality x squared plus 3x is greater than 10 we want to figure out all of the X's that would satisfy this inequality well I encourage you to pause this video now and and I'll give you a hint try to manipulate it the way that you would have if this was a quadratic equation but then as you get to the end try to reason through it because it's the theThis video explains how to solve a quadratic inequality only using a graph. About Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy & Safety How YouTube works Test newTo solve a quadratic inequality, follow these steps: Solve the inequality as though it were an equation. The real solutions to the equation become boundary points for the solution to the inequality. Make the boundary points solid circles if the original inequality includes equality; otherwise, make the boundary points open circles.

Solve Quadratic Inequalities from a Graph Only – YouTube – An equation containing a greater than, greater than or equal to, less than, or less than or equal to sign, where the answers are in the shaded section of the graph either above or below the line. What is a nonlinear system of inequalitiesThe graph of 2 cuts the x-axis at and at "=4. 2. Sketch the graph of the quadratic function. We note the coefficient of . So, the quadratic graph has a minimum point. This is sufficient to make a basic sketch of the quadratic curve. We wish only to know where the curve cuts the horizontal axis. This is shown below. ³So between −2 and +3, the function is less than zero. And that is the region we want, so… x2 − x − 6 < 0 in the interval (−2, 3) Note: x2 − x − 6 > 0 on the interval (−∞,−2) and (3, +∞) And here is the plot of x2 − x − 6: The equation equals zero at −2 and 3. The inequality "<0" is true between −2 and 3.

Solving Quadratic Inequalities – A graph of the quadratic helps us determine the answer to the inequality. We can find the answer graphically by seeing where the graph lies above or below the \(x\) -axis. From the standard form, \(x^2 – 5x + 6\) , \(a>0\) and therefore the graph is a "smile" and has a minimum turning point.Type 1: Algebraically solving x^2\lt a^2. Example: Solve the inequality x^2 \lt 64 When solving quadratic inequalities it is important to remember there are two roots. If the question was solve x^2=a^2 we would simple take the square root of both sides so that x=\pm a.. The range of values that satisfy the inequality is between -8 and 8.This can be expressed as,Quadratic inequalities can have infinitely many solutions, one solution or no solution. We can solve quadratic inequalities graphically by first rewriting the inequality in standard form, with zero on one side. Graph the quadratic function and determine where it is above or below the \(x\)-axis.