e Take the square root of both sides of the given inequality.
F
f Take the cube root of both sides of the given equation.
A
a x=1
B
b x > 3 or x< - 3
C
c 0 ≤ x ≤ 4/3
D
d x=3/4
E
e - 2 ≤ x ≤ 3
F
f x=3
Practice makes perfect
a To solve the given equation, let's first isolate the square root on one side of the equation. We will do this by subtracting 13 from both sides of the equation.
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 0 greater than 3 in the positive direction or in the negative direction.
x > 3 or x< - 3
c We are asked to solve the given inequality.
|3x-2| ≤ 2
To do this, 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 2 away from the midpoint in the positive direction and any number less than or equal to 2 away from the midpoint in the negative direction.
Absolute Value Inequality:& |3x-2| ≤ 2
Compound Inequality:& - 2≤ 3x-2 ≤ 2
We can split this compound inequality into two cases — one where 3x-2 is greater than or equal to -2 and one where 3x-2 is less than or equal to 2.
3x-2 ≥ - 2 and 3x-2 ≤ 2
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 0 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 ≤ 43
Second Solution Set:& 0 ≤ x
Intersecting Solution Set:& 0 ≤ x ≤ 43
d 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 30, which is the lowest common denominator.
To solve the above 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 or equal to 10 away from the midpoint in the positive direction and any number less than 10 away from the midpoint in the negative direction.
Absolute Value Inequality:& |4x-2| ≤ 10
Compound Inequality:& - 10≤ 4x-2 ≤ 10
We can split this compound inequality into two cases, one where 4x-2 is greater than or equal to -10 and one where 4x-2 is less than or equal to 10.
4x-2 ≥ - 10 and 4x-2 ≤ 10
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 - 2 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 ≤ 3
Second Solution Set:& - 2 ≤ x
Intersecting Solution Set:& - 2 ≤ x ≤ 3
f We want to solve the given cubic equation. To do this, we will take the cube root of both sides of the given equation.
