Mister Exam
Other calculators
- Square Inequality Step-by-Step
- Inequality with the module Step by Step
- Logarithmic inequality online
-
How to use it?
- Inequation:
- (x^2+x-20)/(x^2-6*x+y)>0
- (x-5)*(sqrt(9-x))>0
- 3^(log(x-1/x+2))>1/9
- log(x)/log(3)<11-x
-
Identical expressions
- two ,8x^ two + six , two 2x- six , five < four , five - one ,78x- zero ,2x^2
- 2,8x squared plus 6,22x minus 6,5 less than 4,5 minus 1,78x minus 0,2x squared
- two ,8x to the power of two plus six , two 2x minus six , five less than four , five minus one ,78x minus zero ,2x squared
- 2,8x2+6,22x-6,5<4,5-1,78x-0,2x2
- 2,8x²+6,22x-6,5<4,5-1,78x-0,2x²
- 2,8x to the power of 2+6,22x-6,5<4,5-1,78x-0,2x to the power of 2
-
Similar expressions
- 2,8x^2-6,22x-6,5<4,5-1,78x-0,2x^2
- 2,8x^2+6,22x-6,5<4,5-1,78x+0,2x^2
- 2,8x^2+6,22x+6,5<4,5-1,78x-0,2x^2
- 2,8x^2+6,22x-6,5<4,5+1,78x-0,2x^2
2,8x^2+6,22x-6,5<4,5-1,78x-0,2x^2 inequation
The solution
You have entered
[src]
2 2 14*x 311*x 9 89*x x ----- + ----- - 13/2 < - - ---- - -- 5 50 2 50 5
$$\frac{14 x^{2}}{5} + \frac{311 x}{50} - \frac{13}{2} < - \frac{x^{2}}{5} - \frac{89 x}{50} + \frac{9}{2}$$
14*x^2/5 + 311*x/50 - 1*13/2 < -x^2/5 - 89*x/50 + 9/2
Detail solution
Given the inequality:
$$\frac{14 x^{2}}{5} + \frac{311 x}{50} - \frac{13}{2} < - \frac{x^{2}}{5} - \frac{89 x}{50} + \frac{9}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{14 x^{2}}{5} + \frac{311 x}{50} - \frac{13}{2} = - \frac{x^{2}}{5} - \frac{89 x}{50} + \frac{9}{2}$$
Solve:
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$\frac{14 x^{2}}{5} + \frac{311 x}{50} - \frac{13}{2} = - \frac{x^{2}}{5} - \frac{89 x}{50} + \frac{9}{2}$$
to
$$\left(\frac{x^{2}}{5} + \frac{89 x}{50} - \frac{9}{2}\right) + \left(\frac{14 x^{2}}{5} + \frac{311 x}{50} - \frac{13}{2}\right) = 0$$
Expand the expression in the equation
$$\left(\frac{x^{2}}{5} + \frac{89 x}{50} - \frac{9}{2}\right) + \left(\frac{14 x^{2}}{5} + \frac{311 x}{50} - \frac{13}{2}\right) = 0$$
We get the quadratic equation
$$3 x^{2} + 8 x - 11 = 0$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 3$$
$$b = 8$$
$$c = -11$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = 1$$
Simplify
$$x_{2} = - \frac{11}{3}$$
Simplify
$$x_{1} = 1$$
$$x_{2} = - \frac{11}{3}$$
$$x_{1} = 1$$
$$x_{2} = - \frac{11}{3}$$
This roots
$$x_{2} = - \frac{11}{3}$$
$$x_{1} = 1$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$- \frac{11}{3} - \frac{1}{10}$$
=
$$- \frac{113}{30}$$
substitute to the expression
$$\frac{14 x^{2}}{5} + \frac{311 x}{50} - \frac{13}{2} < - \frac{x^{2}}{5} - \frac{89 x}{50} + \frac{9}{2}$$
$$\frac{311}{50} \left(- \frac{113}{30}\right) - \frac{13}{2} + \frac{14 \left(- \frac{113}{30}\right)^{2}}{5} < - \frac{\left(- \frac{113}{30}\right)^{2}}{5} + \frac{9}{2} - \frac{89}{50} \left(- \frac{113}{30}\right)$$
but
Then
$$x < - \frac{11}{3}$$
no execute
one of the solutions of our inequality is:
$$x > - \frac{11}{3} \wedge x < 1$$
$$\frac{14 x^{2}}{5} + \frac{311 x}{50} - \frac{13}{2} < - \frac{x^{2}}{5} - \frac{89 x}{50} + \frac{9}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{14 x^{2}}{5} + \frac{311 x}{50} - \frac{13}{2} = - \frac{x^{2}}{5} - \frac{89 x}{50} + \frac{9}{2}$$
Solve:
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$\frac{14 x^{2}}{5} + \frac{311 x}{50} - \frac{13}{2} = - \frac{x^{2}}{5} - \frac{89 x}{50} + \frac{9}{2}$$
to
$$\left(\frac{x^{2}}{5} + \frac{89 x}{50} - \frac{9}{2}\right) + \left(\frac{14 x^{2}}{5} + \frac{311 x}{50} - \frac{13}{2}\right) = 0$$
Expand the expression in the equation
$$\left(\frac{x^{2}}{5} + \frac{89 x}{50} - \frac{9}{2}\right) + \left(\frac{14 x^{2}}{5} + \frac{311 x}{50} - \frac{13}{2}\right) = 0$$
We get the quadratic equation
$$3 x^{2} + 8 x - 11 = 0$$
This equation is of the form
a*x^2 + b*x + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 3$$
$$b = 8$$
$$c = -11$$
, then
D = b^2 - 4 * a * c =
(8)^2 - 4 * (3) * (-11) = 196
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = 1$$
Simplify
$$x_{2} = - \frac{11}{3}$$
Simplify
$$x_{1} = 1$$
$$x_{2} = - \frac{11}{3}$$
$$x_{1} = 1$$
$$x_{2} = - \frac{11}{3}$$
This roots
$$x_{2} = - \frac{11}{3}$$
$$x_{1} = 1$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$- \frac{11}{3} - \frac{1}{10}$$
=
$$- \frac{113}{30}$$
substitute to the expression
$$\frac{14 x^{2}}{5} + \frac{311 x}{50} - \frac{13}{2} < - \frac{x^{2}}{5} - \frac{89 x}{50} + \frac{9}{2}$$
$$\frac{311}{50} \left(- \frac{113}{30}\right) - \frac{13}{2} + \frac{14 \left(- \frac{113}{30}\right)^{2}}{5} < - \frac{\left(- \frac{113}{30}\right)^{2}}{5} + \frac{9}{2} - \frac{89}{50} \left(- \frac{113}{30}\right)$$
44087 9413 ----- < ---- 4500 1125
but
44087 9413 ----- > ---- 4500 1125
Then
$$x < - \frac{11}{3}$$
no execute
one of the solutions of our inequality is:
$$x > - \frac{11}{3} \wedge x < 1$$
_____
/ \
-------ο-------ο-------
x_2 x_1
Solving inequality on a graph
Rapid solution 2
[src]
(-11/3, 1)
$$x\ in\ \left(- \frac{11}{3}, 1\right)$$
x in Interval.open(-11/3, 1)
The graph