Polynomial Inequalities

An inequality is classified as a linear polynomial inequality when the unknown variable appears only in polynomial expressions and never in a denominator.

For instance, the following is a numerical linear inequality:

$$ 2x + 1 > x - 4 $$

A linear polynomial inequality may also contain fractional terms, provided that the variable does not appear in the denominator.

$$ 2x + \frac{1}{2} > \frac{x-1}{4} $$

Solving a Linear Polynomial Inequality

Consider the numerical inequality below:

$$ \frac{7x-1}{2} > - \frac{2x+1}{4} $$

Note. It is termed a numerical inequality because all coefficients are real numbers. It is a linear polynomial inequality because the variable appears only in linear expressions in the numerators.

The first step in solving the inequality is to clear the denominators by rewriting both sides with a common denominator.

$$ \frac{7x-1}{2} > - \frac{2x+1}{4} $$

The least common multiple of 2 and 4 is 4.

We apply standard fraction operations, multiplying and dividing the first term by 2.

$$ \frac{7x-1}{2} \cdot \frac{2}{2} > - \frac{2x+1}{4} $$

$$ \frac{2(7x-1)}{2\cdot 2} > - \frac{2x+1}{4} $$

$$ \frac{14x-2}{4} > - \frac{2x+1}{4} $$

Both sides now share the same denominator.

Using the standard equivalence rules for inequalities, we multiply both sides by 4 and simplify.

$$ \frac{14x-2}{4} \cdot 4 > - \frac{2x+1}{4} \cdot 4 $$

$$ 14x-2 > - (2x+1) $$

$$ 14x-2 > -2x - 1 $$

At this point all denominators have been eliminated.

Next, collect all variable terms on the left side and all constants on the right side.

Add 2x to both sides:

$$ 14x - 2 + 2x > -2x + 2x - 1 $$

$$ 16x - 2 > -1 $$

Add 2 to both sides:

$$ 16x - 2 + 2 > -1 + 2 $$

$$ 16x > 1 $$

Divide both sides by 16:

$$ 16x \cdot \frac{1}{16} > 1 \cdot \frac{1}{16} $$

The variable is now isolated:

$$ x > \frac{1}{16} $$

This is the solution to the linear polynomial inequality.

Therefore, the inequality is satisfied for all real values with x>1/16.

And so on.