Mister Exam
−(3x+2)+(x−7)>0 inequation
The solution
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-3*x - 2 + x - 7 > 0
$$\left(- 3 x - 2\right) + \left(x - 7\right) > 0$$
-3*x - 2 + x - 7 > 0
Detail solution
Given the inequality:
$$\left(- 3 x - 2\right) + \left(x - 7\right) > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(- 3 x - 2\right) + \left(x - 7\right) = 0$$
Solve:
Given the linear equation:
Expand brackets in the left part
Looking for similar summands in the left part:
Move free summands (without x)
from left part to right part, we given:
$$- 2 x = 9$$
Divide both parts of the equation by -2
$$x_{1} = - \frac{9}{2}$$
$$x_{1} = - \frac{9}{2}$$
This roots
$$x_{1} = - \frac{9}{2}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{9}{2} + - \frac{1}{10}$$
=
$$- \frac{23}{5}$$
substitute to the expression
$$\left(- 3 x - 2\right) + \left(x - 7\right) > 0$$
$$\left(-7 + - \frac{23}{5}\right) + \left(-2 - \frac{\left(-23\right) 3}{5}\right) > 0$$
the solution of our inequality is:
$$x < - \frac{9}{2}$$
$$\left(- 3 x - 2\right) + \left(x - 7\right) > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(- 3 x - 2\right) + \left(x - 7\right) = 0$$
Solve:
Given the linear equation:
-(3*x+2)+(x-7) = 0
Expand brackets in the left part
-3*x-2+x-7 = 0
Looking for similar summands in the left part:
-9 - 2*x = 0
Move free summands (without x)
from left part to right part, we given:
$$- 2 x = 9$$
Divide both parts of the equation by -2
x = 9 / (-2)
$$x_{1} = - \frac{9}{2}$$
$$x_{1} = - \frac{9}{2}$$
This roots
$$x_{1} = - \frac{9}{2}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{9}{2} + - \frac{1}{10}$$
=
$$- \frac{23}{5}$$
substitute to the expression
$$\left(- 3 x - 2\right) + \left(x - 7\right) > 0$$
$$\left(-7 + - \frac{23}{5}\right) + \left(-2 - \frac{\left(-23\right) 3}{5}\right) > 0$$
1/5 > 0
the solution of our inequality is:
$$x < - \frac{9}{2}$$
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Solving inequality on a graph
Rapid solution
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And(-oo < x, x < -9/2)
$$-\infty < x \wedge x < - \frac{9}{2}$$
(-oo < x)∧(x < -9/2)
Rapid solution 2
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(-oo, -9/2)
$$x\ in\ \left(-\infty, - \frac{9}{2}\right)$$
x in Interval.open(-oo, -9/2)