Mister Exam

# Inequality with the module Step by Step

For example, you have entered (calculator here):
$$- \left|{x - 3}\right| + \left|{x^{2} - 1}\right| > 7$$

### Rapid solution

$$\left(-\infty < x \wedge x < - \frac{3 \sqrt{5}}{2} - \frac{1}{2}\right) \vee \left(x < \infty \wedge - \frac{1}{2} + \frac{3 \sqrt{5}}{2} < x\right)$$ $$x\ in\ \left(-\infty, - \frac{3 \sqrt{5}}{2} - \frac{1}{2}\right) \cup \left(- \frac{1}{2} + \frac{3 \sqrt{5}}{2}, \infty\right)$$

### Detail solution

Given the inequality:
$$- \left|{x - 3}\right| + \left|{x^{2} - 1}\right| > 7$$
To solve this inequality, we must first solve the corresponding equation:
$$- \left|{x - 3}\right| + \left|{x^{2} - 1}\right| = 7$$
Solve:
For every modulo expressions in the equation
allow cases, when this expressions ">=0" or "<0",
solve the resulting equation.

1.
$$x - 3 \geq 0$$
$$x^{2} - 1 \geq 0$$
or
$$3 \leq x \wedge x < \infty$$
we get the equation
$$- (x - 3) + \left(x^{2} - 1\right) - 7 = 0$$
after simplifying we get
$$x^{2} - x - 5 = 0$$
the solution in this interval:
$$x_{1} = \frac{1}{2} - \frac{\sqrt{21}}{2}$$
but x1 not in the inequality interval
$$x_{2} = \frac{1}{2} + \frac{\sqrt{21}}{2}$$
but x2 not in the inequality interval

2.
$$x - 3 \geq 0$$
$$x^{2} - 1 < 0$$
The inequality system has no solutions, see the next condition

3.
$$x - 3 < 0$$
$$x^{2} - 1 \geq 0$$
or
$$\left(1 \leq x \wedge x < 3\right) \vee \left(x \leq -1 \wedge -\infty < x\right)$$
we get the equation
$$- (3 - x) + \left(x^{2} - 1\right) - 7 = 0$$
after simplifying we get
$$x^{2} + x - 11 = 0$$
the solution in this interval:
$$x_{3} = - \frac{1}{2} + \frac{3 \sqrt{5}}{2}$$
$$x_{4} = - \frac{3 \sqrt{5}}{2} - \frac{1}{2}$$

4.
$$x - 3 < 0$$
$$x^{2} - 1 < 0$$
or
$$-1 < x \wedge x < 1$$
we get the equation
$$\left(1 - x^{2}\right) - \left(3 - x\right) - 7 = 0$$
after simplifying we get
$$- x^{2} + x - 9 = 0$$
the solution in this interval:
$$x_{5} = \frac{1}{2} - \frac{\sqrt{35} i}{2}$$
but x5 not in the inequality interval
$$x_{6} = \frac{1}{2} + \frac{\sqrt{35} i}{2}$$
but x6 not in the inequality interval

$$x_{1} = - \frac{1}{2} + \frac{3 \sqrt{5}}{2}$$
$$x_{2} = - \frac{3 \sqrt{5}}{2} - \frac{1}{2}$$
$$x_{1} = - \frac{1}{2} + \frac{3 \sqrt{5}}{2}$$
$$x_{2} = - \frac{3 \sqrt{5}}{2} - \frac{1}{2}$$
This roots
$$x_{2} = - \frac{3 \sqrt{5}}{2} - \frac{1}{2}$$
$$x_{1} = - \frac{1}{2} + \frac{3 \sqrt{5}}{2}$$
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
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$\left(- \frac{3 \sqrt{5}}{2} - \frac{1}{2}\right) + - \frac{1}{10}$$
=
$$- \frac{3 \sqrt{5}}{2} - \frac{3}{5}$$
substitute to the expression
$$- \left|{x - 3}\right| + \left|{x^{2} - 1}\right| > 7$$
$$- \left|{\left(- \frac{3 \sqrt{5}}{2} - \frac{3}{5}\right) - 3}\right| + \left|{-1 + \left(- \frac{3 \sqrt{5}}{2} - \frac{3}{5}\right)^{2}}\right| > 7$$
                    2
/        ___\        ___
23   |3   3*\/ 5 |    3*\/ 5  > 7
- -- + |- + -------|  - -------
5    \5      2   /       2       

one of the solutions of our inequality is:
$$x < - \frac{3 \sqrt{5}}{2} - \frac{1}{2}$$
 _____           _____
\         /
-------ο-------ο-------
x2      x1

Other solutions will get with the changeover to the next point
etc.
$$x < - \frac{3 \sqrt{5}}{2} - \frac{1}{2}$$
$$x > - \frac{1}{2} + \frac{3 \sqrt{5}}{2}$$