Mister Exam
9k−35−14k≥0 inequation
The solution
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9*k - 35 - 14*k >= 0
$$- 14 k + \left(9 k - 35\right) \geq 0$$
-14*k + 9*k - 35 >= 0
Detail solution
Given the inequality:
$$- 14 k + \left(9 k - 35\right) \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$- 14 k + \left(9 k - 35\right) = 0$$
Solve:
Given the linear equation:
Looking for similar summands in the left part:
Move free summands (without k)
from left part to right part, we given:
$$- 5 k = 35$$
Divide both parts of the equation by -5
$$k_{1} = -7$$
$$k_{1} = -7$$
This roots
$$k_{1} = -7$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$k_{0} \leq k_{1}$$
For example, let's take the point
$$k_{0} = k_{1} - \frac{1}{10}$$
=
$$-7 + - \frac{1}{10}$$
=
$$- \frac{71}{10}$$
substitute to the expression
$$- 14 k + \left(9 k - 35\right) \geq 0$$
$$\left(\frac{\left(-71\right) 9}{10} - 35\right) - \frac{\left(-71\right) 14}{10} \geq 0$$
the solution of our inequality is:
$$k \leq -7$$
$$- 14 k + \left(9 k - 35\right) \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$- 14 k + \left(9 k - 35\right) = 0$$
Solve:
Given the linear equation:
9*k-35-14*k = 0
Looking for similar summands in the left part:
-35 - 5*k = 0
Move free summands (without k)
from left part to right part, we given:
$$- 5 k = 35$$
Divide both parts of the equation by -5
k = 35 / (-5)
$$k_{1} = -7$$
$$k_{1} = -7$$
This roots
$$k_{1} = -7$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$k_{0} \leq k_{1}$$
For example, let's take the point
$$k_{0} = k_{1} - \frac{1}{10}$$
=
$$-7 + - \frac{1}{10}$$
=
$$- \frac{71}{10}$$
substitute to the expression
$$- 14 k + \left(9 k - 35\right) \geq 0$$
$$\left(\frac{\left(-71\right) 9}{10} - 35\right) - \frac{\left(-71\right) 14}{10} \geq 0$$
1/2 >= 0
the solution of our inequality is:
$$k \leq -7$$
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