Exponents & Inequalities
Definitions
- Inequality - A comparison of two values or expressions.
For example, 20x < 40 is an inequality whereas x = 2 is an equation.
- Equation - A statement declaring the equality of two expressions.
For example, 4x = 40 is an equation whereas 4x > 40 is an inequality.
- Exponent - The number of times a quantity is multiplied by itself.
For example, in the expression 58, the number 8 is the exponent.
Operating With Inequalities: Exponents
In dealing with inequalities that involve exponents, these inequalities behave much like traditional equations. Inequalities with an even exponent usually have two solutions while inequalities with an odd exponent have one solution.
Odd Exponents
An inequality with an odd exponent behaves exactly like an inequality without an exponent or a traditional equation with an odd exponent. The one word of caution--which also applies to inequalities without exponents--is that you must know the sign of a variable before you can divide or multiply by it (see multiplication & division with inequalities).
x3 < 27
(x3)1/3 < 271/3
x < 3 [it is not necessary to flip the inequality sign since no negative numbers are involved]
As a check, if x = 4 (outside the solution set), x3 = 64, which does not fit the inequality (i.e., 64 is not less than 27). However, if x = 2 (inside the solution set), x3 = 8, which does fit the inequality (i.e., 8 is less than 27).
2x3 + 14 > 30
2x3 + 14 - 14 > 30 - 14
2x3 > 16
x3 > 8
(x3)1/3 > 81/3
x > 2
As a check, if x = 0 (outside the solution set), 2x3 + 14 = 14, which does not fit the inequality (i.e., 14 is not greater than or equal to 30). However, if x = 2 (inside the solution set), 2x3 + 14 = 30, which does fit the inequality (i.e., 30 is greater than or equal to 30).
Even Exponents
As stated above, an inequality with an even exponent typically has two solutions. The reason for this is that x can be either positive or negative. Consequently, when evaluating an even exponent within an inequality, we deal with two cases: x is positive, x is negative.
2x
2 > 32
x
2 > 16
Case 1: x is positive
x
> 4
Case 2: x is negative
x
< -4
Note: The sign of the inequality flipped because we took the root of a negative number.
The solution can be represented graphically.
As a check, if x = -5 (within the solution set), 2x
2 = 50, which fits the inequality. However, if x = -2 (outside the solution set), 2x
2 = 8, which does not fit the inequality.
3x2 < 27
x2 < 9
Case 1: x is positive
x < 3
Case 2: x is negative
x > -3
Multiple Inequalities
Multiple inequalities with exponents are solved just as multiple inequalities without exponents are solved.
If 2x2 + 5 < 13 and x2 < 9, what is the range of possible values for x?
1.) Solve each inequality alone.
2x2 + 5 < 13
2x2 < 8
x2 < 4
x < 2 AND x > -2
x2 < 9
x < 3 AND x > -3
2.) Combine each inequality and find the overlap (i.e., the areas where each inequality is satisfied--this area is the solution).
x < 2
x > -2
x < 3
x > -3
The area of overlap--i.e., the solution to the set of inequalities--is where x < 2 and x > -2
For many students, the above set of inequalities can best be understood graphically. The solution to the set of inequalities is the overlapping graphical area.