Mister Exam
7^(x²+5x)>1 inequation
The solution
Detail solution
Given the inequality:
$$7^{x^{2} + 5 x} > 1$$
To solve this inequality, we must first solve the corresponding equation:
$$7^{x^{2} + 5 x} = 1$$
Solve:
$$x_{1} = -5$$
$$x_{2} = 0$$
$$x_{1} = -5$$
$$x_{2} = 0$$
This roots
$$x_{1} = -5$$
$$x_{2} = 0$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-5 + - \frac{1}{10}$$
=
$$- \frac{51}{10}$$
substitute to the expression
$$7^{x^{2} + 5 x} > 1$$
$$7^{\frac{\left(-51\right) 5}{10} + \left(- \frac{51}{10}\right)^{2}} > 1$$
one of the solutions of our inequality is:
$$x < -5$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < -5$$
$$x > 0$$
$$7^{x^{2} + 5 x} > 1$$
To solve this inequality, we must first solve the corresponding equation:
$$7^{x^{2} + 5 x} = 1$$
Solve:
$$x_{1} = -5$$
$$x_{2} = 0$$
$$x_{1} = -5$$
$$x_{2} = 0$$
This roots
$$x_{1} = -5$$
$$x_{2} = 0$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-5 + - \frac{1}{10}$$
=
$$- \frac{51}{10}$$
substitute to the expression
$$7^{x^{2} + 5 x} > 1$$
$$7^{\frac{\left(-51\right) 5}{10} + \left(- \frac{51}{10}\right)^{2}} > 1$$
51
---
100 > 1
7
one of the solutions of our inequality is:
$$x < -5$$
_____ _____
\ /
-------ο-------ο-------
x1 x2Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < -5$$
$$x > 0$$
Solving inequality on a graph
Rapid solution 2
[src]
(-oo, -5) U (0, oo)
$$x\ in\ \left(-\infty, -5\right) \cup \left(0, \infty\right)$$
x in Union(Interval.open(-oo, -5), Interval.open(0, oo))
Rapid solution
[src]
Or(And(-oo < x, x < -5), 0 < x)
$$\left(-\infty < x \wedge x < -5\right) \vee 0 < x$$
(0 < x)∨((-oo < x)∧(x < -5))