What is the answer to [x+5] = [x-3]?

Question

The [ ] 's are absolute values.
Also, what is the answer to:
[0.5x+3] = x2 - 4
The 2 beside the x is x squared.
I would greatly appreciate it if you could show all the work as best as you could!
Thanks!

Answer 1

x = -1 is the answer.

Answer 2

In the first equation, x= -1.
This is because you can tell that 5 and 3 are close in number. There is a difference of two and one might be a good number that could make them equal (add one to 3, subtract one from 5 for example).

Since this equation involves absolute value, you can plug in 1 and -1. Only -1 solves the equation because when
x=1 l 1+5 l does not equal l 1-3 l. (l l=absolute value).
x= -1 l -1+5 l equals to l -1-3 l, simplified equals l 4 l = l -4 l

For the second problem, you know that absolute value means that whether the value is positive or negative, you write the positive value because absolute value is the positive distance from zero. You have to break this into two equations in order to solve for values of x.

0.5x+3=x2 - 4 and 0.5x+3= -x2 +4
The answer to one equation remains the same while the other is negated in order to take the absolute value into consideration.

Next, bring the equation all over onto one side. It doesn't matter which but I did it one way and ended up with:
x^2-0.5x-7=0 and -x^2-0.5x+7=0

Since these equations can't be solved using factoring by hand, (FOIL, first, outer, inner, last), you need to use the quadratic equation to solve both equations.
Quadratic Equation= -b + or - √(b^2-4ac)
2a
Supposed to be a fraction with 2a on the bottom.

Find values for A, B, and C according to the form Ax^2+Bx+C=0 and then plug them into the quadratic equation to solve.

For the first equation I gave you, A=1, B= -0.5, and C= -7.
[-(-0.5) + or - √((-0.5)^2 - 4(1 x -7))] / (2 x 1)
[0.5 + or - √(28.25)] / (2)
You can punch this in the calculator and you should get about 2.9075... and -2.4075... for the values of x in this equation.

For the second equation I gave you, A= -1, -0.5, and C= 1.
[-(-0.5) + or - √((-0.5)^2 - 4(-1 x 1))] / (2 x -1)
[0.5 + or - √(4.25)] / (-2)
After punching this into the calculator, you should get about 0.78077... and -1.28077... for the values of x in this equation.

To check your answers plug them (the exact values) back into their original equation and you should get zero. I checked it on my calculator so they should be correct.

Answer 3

1.)
|x + 5| = |x - 3|

To solve this, you need to set up solutions for each value range of x. Essentially, the equation of the line changes at the point where the equation inside the absolute value marks equals 0. When the line changes, multiply the equation by -1.

Where 3 >= x >= -5: x + 5 = - x + 3
2x = -2
x = -1 (solution)

Where x > 3: x + 5 = x - 3
0 = 8 (no solutions)

Where x < = -5: - x - 5 = - x + 3
0 = 8 (no solutions)

2.) [0.5x+3] = x^2 - 4
Where x >= -6: .5x + 3 = x^2 - 4
-x^2 + .5x + 7 = 0
(where a quadratic eqn = ax^2 + bx + c)
x = (-b +/- (b^2 -4ac)^1/2)/2a
x = (-.5 +/- (.25 - (4 * 7 * -1))^1/2)/-2)
x = (-.5 +/- (5.315))/-2) (note: 5.315 = sqrt 28.25)
x = 2.4075, -2.9075 (solution 1 and 2)

Where x < -6: -.5x - 3 = x^2 - 4
x^2 + .5x - 1 = 0
x = (-b +/- (b^2 -4ac)^1/2)/2a
x = (-.5 +/- (.25 - (4 * 1 * -1))^1/2)/2)
x = (-.5 +/- (2.062))/2) (2.062 = sqrt 4.25)
x = .781, -1.281 (ignore, because x must be <-6 for this equation).

Answer 4

First question:

[x+5] = [x-3]
i swear something is missing, but by absolute values you mean the modulus (i,e, +ve and -ve not concerned)

Then x= -1


Second question

[0.5x+3] = (x^2) - 4

i cant figure it out, something wrong with the notation, something missed out?

Answer 5

1. no clue

2.
[1/2x+3]=x2-4
1/2x-x2= -7
x2-1/2x=7
x= -3.5 and the square root of 7

By the way, are you graphing parabolas?