Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation 2+3x=-2x-13
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Equation cos(x)=-(1/2)
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Equation |x^2-1|-|x-3|=7
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Equation 2*x^2+6*x+9=0
- Express {x} in terms of y where:
- -14*x-16*y=14
- 16*x-17*y=-9
- -14*x-3*y=-5
- -16*x-20*y=6
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Identical expressions
- |x^ two - one |-|x- three |= seven
- module of x squared minus 1| minus |x minus 3| equally 7
- module of x to the power of two minus one | minus |x minus three | equally seven
- |x2-1|-|x-3|=7
- |x²-1|-|x-3|=7
- |x to the power of 2-1|-|x-3|=7
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Similar expressions
- |x^2-1|-|x+3|=7
- |x^2-1|+|x-3|=7
- |x^2+1|-|x-3|=7
|x^2-1|-|x-3|=7 equation
The teacher will be very surprised to see your correct solution 😉
The solution
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[src]
| 2 | |x - 1| - |x - 3| = 7
$$- \left|{x - 3}\right| + \left|{x^{2} - 1}\right| = 7$$
Detail solution
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
The final answer:
$$x_{1} = - \frac{1}{2} + \frac{3 \sqrt{5}}{2}$$
$$x_{2} = - \frac{3 \sqrt{5}}{2} - \frac{1}{2}$$
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
The final answer:
$$x_{1} = - \frac{1}{2} + \frac{3 \sqrt{5}}{2}$$
$$x_{2} = - \frac{3 \sqrt{5}}{2} - \frac{1}{2}$$
Rapid solution
[src]
___
1 3*\/ 5
x1 = - - + -------
2 2
$$x_{1} = - \frac{1}{2} + \frac{3 \sqrt{5}}{2}$$
___
1 3*\/ 5
x2 = - - - -------
2 2
$$x_{2} = - \frac{3 \sqrt{5}}{2} - \frac{1}{2}$$
x2 = -3*sqrt(5)/2 - 1/2
Sum and product of roots
[src]
sum
___ ___ 1 3*\/ 5 1 3*\/ 5 - - + ------- + - - - ------- 2 2 2 2
$$\left(- \frac{3 \sqrt{5}}{2} - \frac{1}{2}\right) + \left(- \frac{1}{2} + \frac{3 \sqrt{5}}{2}\right)$$
=
-1
$$-1$$
product
/ ___\ / ___\ | 1 3*\/ 5 | | 1 3*\/ 5 | |- - + -------|*|- - - -------| \ 2 2 / \ 2 2 /
$$\left(- \frac{1}{2} + \frac{3 \sqrt{5}}{2}\right) \left(- \frac{3 \sqrt{5}}{2} - \frac{1}{2}\right)$$
=
-11
$$-11$$
-11
The graph