Quadratic Inequalities 1

Types of lines used graphing inequalities-Dotted line

< or > used in the dotted line,and the number is not included,plus remaining of the points are included.

Types of lines used graphing inequalities-Solid Line

< or equal or > or equal used in the solid line,and the point is also included,and remaining of the points.

A graph of quadratic function y = f(x) is shown below.

A

What is a nonlinear system of inequalities

A collection of equations and inequalities where at least one of the inequalities is nonlinear.

What is a nonlinear system

A system of equations that contains at least one non-linear equation.

What is an inequality

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.

Inequality

An inequality is type of equation but it has symbols like greater than,or less than.

A small rock falling from the top of a 124-ft-tall building with an initial downward velocity of -30 ft/sec is modeled by the equation h(t) = -16t2 - 30t + 124, where t is the time in seconds. For which interval of time does the rock remain in the air?

C T<2

Line when graphing < or >

Dotted line

Comparison between inequalities and equations (Visual)

Equation: y=-2(x-6)^2+4 Inequality: y<-2(x-6)^2+4 Equations have an equal sign and answers are only on the parabola. Inequalities have a greater than, greater than or equal to, less than, or less than or equal to sign, and the possible answers are in the shaded region.

Comparison of systems of equation solutions to systems of inequalities solutions-visual comparison

Equations-equal sign,Solve the system to get the x and y value. Inequalities-have greater or equal,less or equal,find the intersection means discovering the solution of two lines. The similarities of both is the solution,and they both get the x and y value.

Comparison between systems of equations and systems of inequalities (Visual)

Equations: y= x^2-2x+3 y=2x+4 Inequalities: y≥3x^2+2 y<2x+6 The only possible answers for the systems of equations are the two set intersections, while the possible answers for the systems of inequalities are in the range where both equations' shaded areas are overlapping.

Comparison of equations to inequalities-visual comparison

Equations:equal sign,easier to solve,get the one solution. Inequalities:greater or equal,lesser or equal,and when dividing the negative number,you switch the sign. Similarities:get the solution.

How do you find the solution to a nonlinear system of inequalities Ex. y> 2x^2+3x y≤ 0.5x^2+10

First, you correctly graph the ineqalities on the graph, dotting or not dotting the line depending on the symbol used. Then, you shade in the correct regions of each inequality. The answer is the overlapping shaded regions. Ex. First you graph y> 2x^2+3x and y≤ 0.5x^2+10 on the coordinate plane. Then, you shade inside the parabola for y> 2x^2+3x, and outside the parabola for y≤ 0.5x^2+10. The area where the shaded regions of y> 2x^2+3x and y≤ 0.5x^2+10 overlap is the answer.

Shaded outside the inequality

If the test point inside the equality is false, or if the test point outside of the inequality is true

Shaded inside the inequality

If the test point inside the equality is true, or if the test point outside of the inequality is false

Non-linear system

Non-linear system is a system of equations that have one non-linear equation. The example of non-linear were parabolas,circles,radical functions.

Non-linear system of inequalities

Non-linear system of inequalities have one solution,and which derived by the feasible region,that satisfies the equations. Non-linear systems of inequalities have no equal symbol,and don't the straight line. Instead,it will have the parabolas,circles,etc.

Reyna runs a textile company that manufactures T-shirts. The profit, p, made by the company is modeled by the function mc025-1.jpg, where s is the number of T-shirts sold. How many T-shirts should be sold to earn a profit of more than $2,000?

S>42 B

Line when graphing ≤ or ≥

Solid line

[-2, -1/2]

Solve 2x²+5x≤-2

(-∞, -1/2)∪(-1/2, ∞)

Solve 4x²+4x+1>0

(-∞ , −2]∪ [2, ∞)

Solve 4x²-16≥0

(-3/2, 3/2)

Solve 4x²<9

[-1, 5/2]

Solve 5-2x²≥-3x

(-∞, 2/5]∪[5,∞)

Solve 5x²+10x≥27x

(1/5, 1)

Solve 5x²-6x<-1

(-∞, -7/2)∪(2/3, ∞)

Solve 6x²>14-17x

(−4, 1)

Solve 7x²+21x−28<0

[-3, -4/9]

Solve 9x²+31x≤-12

[-1/3]

Solve 9x²+6x+1≤0

[-3, -1]

Solve x²+4x+3≤0

(-∞, -2)∪(2, ∞)

Solve x²-4>0

(-∞, 2)∪(4, ∞)

Solve x²-6x+8>0

(-2, 8)

Solve x²-6x<16

[0, 6]

Solve x²-6x≤0

Comparison between inequalities and equations (Words)

The difference between equations and inequalities is that equations have answers only along the parabola, lines, ect., while inequalities have answers inside the shaded region near the parabola, lines, ect.

Comparison between systems of equations and systems of inequalities (Words)

The difference between systems of equations and systems of inequalities is that in systems of equations, the answers can only be intersections and there are fewer and only set and exact answers, while in systems of inequalities, the answers are in the shaded region of the intersecting parabolas, lines, ect. and there are more possible answers.

Visual Example of a Inequality graph of line use and shading.

The example shown right is the example of the inequality graph. It use the dotted line because the coordinate is not included,and remaining of them is included,if you plug into the equation. Let's take the example,like the equation for this graph is y<x-2. To do the shading,I take the test point (0,0),and I plug 0 into the equation 0<0-2,which results 0<-2. Since,it is a false statement,you have to shade,where it can't be a zero.

Visual Example of system of inequalities of line use and shading.

The picture on the right is the example of systems of inequalities. Since the picture gave the equations,y>-3x+5 and y< or equal to x-2. The y>-3x+5,you have to draw the dotted line. To do this,the line starts from y-intercept,which is (0,5),and move down 3 over 1. Since,they gave the equation,and want to see where to shade,you always want to have the test point to shade,and it is (0,0). Then I plug into the equation,which results 0>-3(0)+5,and you should get 0>5. Since,this is a false statement,you have to shade the side,where there is no (0,0). Repeat the same process above,and the second equation is y< or equal x-2. In this,you have to use the solid line,because the coordinate is also included. To draw the line,you use the y-intercept,and move the slope. The The line starts from (0,-2),and then move up 1. Then you have the solid line. For shading,you use the test point (0,0) from the equation y< or equal x-2,and I plug in and results 0< or equal 0-2,and the answer should come with 0< or equal to -2. Since it is the false statement,then you have to shade,where there no (0,0). The solution for this type of system of inequalities can get by the feasible region,where both lines meet,and you see that the most feasible region is at the right side. The best to do this by graphing method to get the feasible region.

Why you shade

The reason we shade,because that determines that the points were included. For example,if you plug an equation and shade,and want to test it out. Typically,you choose the point from shaded part,and test it,to see if they are equal.

Reason for shading

The shaded region contains possible answers for the inequality

How do you get the solution by the Non-linear system of inequalities

To get the solution from the Non-linear system of inequalities by the graphing method. If you graph the inequalities,then you can see the feasible region or the most shading region from the both of the non-linear system of inequalities. If you have most feasible region from both of the inequalities,then it will satisfy the both of the equations.

How you shade

We shade based out of the equation. For example,y> and equal to 2x-4. First,you plot the point based out of the equation,and to shade,you choose the test point. Typically,I recommend to choose (0,0) because it is easier,and I plug the point into the equation,0 greater or equal to 2(0)-4,and then you solve,and it results 0> or equal to -4. Since,0 is greater to -4,then you shade the part that belongs to the zero.

Comparison of equations to inequalities-words

equal sign,greater or equal,lesser or equal.

What are the solutions to the inequality mc003-1.jpg

mc003-4.jpg C

What is the solution set of the quadratic inequality 4(x + 2)2 <0?

mc009-2.jpg {x/x=-2} A

Which quadratic inequality does the graph below represent?

mc011-2.jpg A y<2x^2-8x+3

The height of a triangle is 4 in. greater than twice its base. The area of the triangle is no more than 168 in.2. Which inequality can be used to find the possible lengths, x, of the base of the triangle?

mc015-2.jpg B x(x+2)<168

Greg is in a car at the top of a roller-coaster ride. The distance, d, of the car from the ground as the car descends is determined by the equation d = 144 - 16t2, where t is the number of seconds it takes the car to travel down to each point on the ride. For which interval of time is Greg's car moving in the air?

mc020-4.jpg D

Comparison of systems of equations solutions to systems of inequalities solutions

solve the system to get the x and y value,intersection means discovering the solution of two lines.

Jerald jumped from a bungee tower. If the equation that models his height, in feet, is h = -16t2 + 729, where t is the time in seconds, for which interval of time is he within 104 feet above the ground?

t > 6.25 A

The graph below represents the solution set of which inequality?

x^2 + 2x - 8 < 0 B