Power Inequalities
Power inequalities are inequalities of the form $$ ax^n + b > 0 $$ or $$ ax^n + b < 0 $$ where the coefficient a is a positive integer, a>0. These expressions are typically straightforward to analyze because once the variable term is isolated, the inequality reduces to evaluating an appropriate power or root.
The solution method is simple and systematic.
We isolate the term containing the variable and then apply standard algebraic operations to interpret the resulting inequality.
A practical example
Consider the inequality:
$$ x^3 - 8 > 0 $$
Use the invariance property by adding 8 to both sides:
$$ x^3 - 8 + 8 > 0 + 8 $$
This produces a cleaner expression involving the variable:
$$ x^3 > 8 $$
Now apply the cube root to both sides:
$$ \sqrt[3]{x^3} > \sqrt[3]{8} $$
$$ \sqrt[3]{x^3} > \sqrt[3]{2^3} $$
The cube root of 8 is 2, so:
$$ x > 2 $$
Hence, the inequality holds for all real numbers in the open interval (2,+∞).
Note. A cube root has odd index and therefore yields a single real result. $$ 2\cdot 2\cdot 2 = 8 $$ By contrast, $$ (-2)\cdot(-2)\cdot(-2) = -8 $$ so the negative value does not satisfy the condition.
Example 2
Now consider the inequality:
$$ x^4 - 81 < 0 $$
Isolate the power term:
$$ x^4 - 81 + 81 < 0 + 81 $$
$$ x^4 < 81 $$
Apply the fourth root to both sides:
$$ \sqrt[4]{x^4} < \sqrt[4]{81} $$
$$ x < \sqrt[4]{81} $$
Since 81 is equal to 34:
$$ x < \sqrt[4]{3^4} $$
A fourth root has even index, so it produces two real boundary values, +3 and -3.
$$ \begin{cases} x < +3 \\ \\ x > -3 \end{cases} $$
The inequality is therefore satisfied for all real numbers in the open interval (-3,3):
$$ -3 < x < 3 $$
Note. An even-indexed root yields two symmetric real values. $$ 3\cdot 3\cdot 3\cdot 3 = 81 $$ $$ (-3)\cdot(-3)\cdot(-3)\cdot(-3) = 81 $$ These values define the endpoints of the interval in which the inequality is valid.
The same approach applies to more complex expressions of this type.
