Mister Exam
2x/3-x+1/4<=0 inequation
The solution
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2*x 1 --- - x + - <= 0 3 4
$$\left(- x + \frac{2 x}{3}\right) + \frac{1}{4} \leq 0$$
-x + (2*x)/3 + 1/4 <= 0
Detail solution
Given the inequality:
$$\left(- x + \frac{2 x}{3}\right) + \frac{1}{4} \leq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(- x + \frac{2 x}{3}\right) + \frac{1}{4} = 0$$
Solve:
Given the linear equation:
Looking for similar summands in the left part:
Move free summands (without x)
from left part to right part, we given:
$$- \frac{x}{3} = - \frac{1}{4}$$
Divide both parts of the equation by -1/3
$$x_{1} = \frac{3}{4}$$
$$x_{1} = \frac{3}{4}$$
This roots
$$x_{1} = \frac{3}{4}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{3}{4}$$
=
$$\frac{13}{20}$$
substitute to the expression
$$\left(- x + \frac{2 x}{3}\right) + \frac{1}{4} \leq 0$$
$$\left(- \frac{13}{20} + \frac{\frac{13}{20} \cdot 2}{3}\right) + \frac{1}{4} \leq 0$$
but
Then
$$x \leq \frac{3}{4}$$
no execute
the solution of our inequality is:
$$x \geq \frac{3}{4}$$
$$\left(- x + \frac{2 x}{3}\right) + \frac{1}{4} \leq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(- x + \frac{2 x}{3}\right) + \frac{1}{4} = 0$$
Solve:
Given the linear equation:
2*x/3-x+1/4 = 0
Looking for similar summands in the left part:
1/4 - x/3 = 0
Move free summands (without x)
from left part to right part, we given:
$$- \frac{x}{3} = - \frac{1}{4}$$
Divide both parts of the equation by -1/3
x = -1/4 / (-1/3)
$$x_{1} = \frac{3}{4}$$
$$x_{1} = \frac{3}{4}$$
This roots
$$x_{1} = \frac{3}{4}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{3}{4}$$
=
$$\frac{13}{20}$$
substitute to the expression
$$\left(- x + \frac{2 x}{3}\right) + \frac{1}{4} \leq 0$$
$$\left(- \frac{13}{20} + \frac{\frac{13}{20} \cdot 2}{3}\right) + \frac{1}{4} \leq 0$$
1/30 <= 0
but
1/30 >= 0
Then
$$x \leq \frac{3}{4}$$
no execute
the solution of our inequality is:
$$x \geq \frac{3}{4}$$
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Solving inequality on a graph
Rapid solution
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And(3/4 <= x, x < oo)
$$\frac{3}{4} \leq x \wedge x < \infty$$
(3/4 <= x)∧(x < oo)