c Take the cube root of both sides of the given equation.
A
a x=1
B
b x<- 1 or x>7
C
c x=7
Practice makes perfect
a To solve the given quadratic equation, we will first rewrite the equation so all of the terms are on the left-hand side and then simplify as much as possible. Let's begin by using distribution.
Now, look closely at the equation above. It would be much easier to solve if the expression on the left-hand side of the equation was a perfect square trinomial. To determine if an expression is a perfect square trinomial, we need to ask ourselves three questions.
Is the first term a perfect square?
x^2= x^2 âś“
Is the last term a perfect square?
1= 1^2 âś“
Is the middle term twice the product of 1 and x?
2x=2* 1* x âś“
As we can see, the answer to all three questions is yes! Therefore, we can write the trinomial as the square of a binomial. Note there is a subtraction sign in the middle.
x^2-2x+1=0 ⇔ ( x- 1)^2=0
To solve the above quadratic equation, we will take the square root of both sides of the equation. Since this method gives two solutions — a negative and a positive — remember to consider them both by adding ± to the solution.
To solve the above inequality, we will create a compound inequality by removing the absolute value. In this case the solution set contains the numbers that make the distance between x and 3 greater than 4 in the positive direction or in the negative direction.
x-3 > 4 or x-3< - 4
Let's isolate x in both of these cases before writing the solution set.
This inequality tells us that all values less than - 1 will satisfy the inequality.
Solution Set
The solution to this type of compound inequality is the combination of the solution sets.
First Solution Set:& x>7
Second Solution Set:& x<- 1
Combined Solution Set:& x<- 1 or x>7
This inequality tells us that all values greater than 7 or less than - 1 will satisfy the inequality. Now, we can check our solution by substituting four values of x into the original inequality.
Value of x
Substitute
Simplify
- 2
10| - 2-3|? > 40
50 > 40
0
10| 0-3|? > 40
30 ≯ 40
6
10| 6-3| ? > 40
30 ≯ 40
8
10| 8-3| ? > 40
50 > 40
As we can see, values of x greater than 7 or less than - 1 are indeed the solution of the given inequality.
c We want to solve the given cubic equation. To do this, we will take the cube root of both sides of the given equation.
