a This equation would be much easier to solve if it had no fractions. We can start solving by changing this equation to a simpler equivalent equation by eliminating fractions. To do this we will multiply both sides of the equation by 12, which is the lowest common denominator.
To solve an equation we should first gather all of the variable terms on one side of the equation and all of the constant terms on the other side using the Properties of Equality.
To solve this inequality we will create a compound inequality by removing the absolute value. In this case, the solution set is any number less than sqrt(12) away from the midpoint in the positive direction and any number less than sqrt(12) away from the midpoint in the negative direction.
&Absolute value Compound [-0.5em]
& inequality: inequality:
&|b-4| < sqrt(12) - sqrt(12)< b-4 < sqrt(12)
We can split this compound inequality into two cases, one where b-4 is greater than -sqrt(12) and one where b-4 is less than sqrt(12).
b-4 > - sqrt(12) and b-4 < sqrt(12)
Let's isolate b in both of these cases before writing the solution set.
This inequality tells us that all values greater than 4-sqrt(12) will satisfy the inequality.
Solution Set
The solution to this type of compound inequality is the overlap of the solution sets. Let's recombine our cases back into one compound inequality.
First solution set:& b<4+sqrt(12)
Second solution set:& 4-sqrt(12)
c To solve the given inequality let's first isolate the absolute value on one side. We will do this by adding 9 to both sides of the inequality.
We will create a compound inequality by removing the absolute value. In this case, the solution set is any number less than or equal to 30 away from the midpoint in the positive direction and any number less than or equal to 30 away from the midpoint in the negative direction.
Absolute Value Inequality:& |3+x| ≤ 30
Compound Inequality:& - 30≤ 3+x ≤ 30
We can split this compound inequality into two cases — one where 3+x is greater than or equal to -30 and one where 3+x is less than or equal to 30.
3+x ≥ - 30 and 3+x ≤ 30
Let's isolate x in both of these cases before writing the solution set.
This inequality tells us that all values greater than or equal to - 33 will satisfy the inequality.
Solution Set
The solution to this type of compound inequality is the overlap of the solution sets. Let's recombine our cases back into one compound inequality.
First Solution Set:& x ≤ 27
Second Solution Set:& - 33 ≤ x
Intersecting Solution Set:& - 33 ≤ x ≤ 27
an^2+ bn+ c=0 ⇕ n=- b± sqrt(b^2-4 a c)/2 a
We first need to identify the values of a, b, and c.
5n^2-11n+2=0 ⇕ 5n^2+( - 11)n+ 2=0
We see that a= 5, b= - 11, and c= 2. Let's substitute these values into the Quadratic Formula.
