6.7 Rational inequalities

Ideas in this lesson

Find critical values

Numerator zeros and denominator zeros divide the number line into intervals for sign testing.

Build a sign chart

Test a value in each interval between critical values. The sign is constant on an interval without a zero or pole.

Choose endpoints correctly

A non-strict inequality can include a numerator zero. A denominator zero is always excluded.

Write the solution set

Select intervals with the required sign and use interval notation with correct endpoint inclusion.

Move a nonzero comparison to one side, combine fractions, and explain why multiplying by an unknown-sign denominator is unsafe

move a nonzero comparison to one side, combine fractions, and explain why multiplying by an unknown-sign denominator is unsafe.

Solve with tables and graphs and validate boundaries algebraically rather than infer exact completeness from a sparse table

solve with tables and graphs and validate boundaries algebraically rather than infer exact completeness from a sparse table.

Create contextual rational inequalities and interpret integer/positive-domain thresholds, including strict versus non-strict counts

create contextual rational inequalities and interpret integer/positive-domain thresholds, including strict versus non-strict counts.

Solve multi-factor/repeated-factor rational inequalities while retaining canceled-factor holes and testing signs that need not alternate

solve multi-factor/repeated-factor rational inequalities while retaining canceled-factor holes and testing signs that need not alternate.