An absolute value measures an expression's distance from a midpoint on a number line.
|x-3| = 2
This equation means that the distance is 2, either in the positive direction or the negative direction.
|x-3|= 2 ⇒ lx-3= 2 x-3= - 2
To find the solutions to the absolute value equation, we need to solve both of these cases for x.
Both 5 and 1 are solutions to the absolute value equation.
b To solve the given quadratic equation, we can isolate the variable expression raised to the power of 2 and take the square roots. Remember to consider the positive and negative solutions.
We found that x=± 2 + 2. We will now find both solutions by using the positive and negative signs.
x=± 2 + 2
x=2 + 2
x=-2 + 2
x=4
x=0
There are two solutions to the given equation, x=4 and x=0.
c To solve equations with a variable expression inside a radical, we first want to make sure the radical is isolated. Then we can raise both sides of the equation to a power equal to the index of the radical. Note that the square root in the given equation is already isolated.
Note that we obtained a quadratic equation. To solve it we will use the Quadratic Formula.
ax^2+ bx+ c=0 ⇕ x=- b± sqrt(b^2-4 a c)/2 aWe first need to identify the values of a, b, and c.
x^2-5x-14=0 ⇕ 1x^2+( - 5)x+( -14)=0
We see that a= 1, b= - 5, and c= -14. Let's substitute these values into the Quadratic Formula.
Using the Quadratic Formula, we found that the solutions of the given equation are x= 5± 92.
x=5± 9/2
x_1=5+9/2
x_2=5-9/2
x_1=14/2
x_2=-4/2
x_1= 7
x_2= -2
The solutions are x_1=7 and x_2= -2. Finally, we will check for extraneous solutions by substituting both x= 7 and x= -2 into the original equation. Let's start with x= 7.
This time we obtained a false statement. Therefore, x= -2 is not a solution to the original equation.
d An absolute value measures an expression's distance from a midpoint on a number line.
|2x+5|= 3x+4
This equation means that the distance is 3x+4, either in the positive direction or the negative direction.
|2x+5|= 3x+4
⇓
l2x+5= (3x+4) 2x+5= -(3x+4)
To find the solutions to the absolute value equation, we need to solve both of these cases for n.
Both x=1 and x=- 95 are solutions to the equation. Finally, we will check for extraneous solutions by substituting both x=1 and x=- 95 into the original equation. Let's start with x=1.
