Quadratic inequalities

Order the roots a < b. The product (x − a)(x − b) is positive outside the roots, negative between them and zero at the roots. Strict > and < exclude roots; ≤ and ≥ include them. Select the required intervals.

Order the roots a < b. The product (x − a)(x − b) is positive outside the roots, negative between them and zero at the roots. Strict > and < exclude roots; ≤ and ≥ include them. Select the required intervals.

For c > 0, x² < c means −√c < x < √c. For x² > c, take x < −√c or x > √c. Strict inequalities exclude the boundary values.

In the strict double inequality a < x < b, x must be greater than a and less than b. The boundary values themselves are excluded.

1. Find the smallest integer strictly greater than the lower bound.
2. Find the largest integer strictly less than the upper bound.
3. Count all integers from the first to the last, inclusive: last − first + 1. If the first exceeds the last, there are no solutions.

When both bounds a and b are integers and a < b, the permitted integers run from a + 1 to b − 1, so their number is b − a − 1. Remember that 0 is also an integer and must be counted if it lies inside the interval.

Order the roots a < b. The product (x − a)(x − b) is positive outside the roots, negative between them and zero at the roots. Strict > and < exclude roots; ≤ and ≥ include them. Select the required intervals.

Order the roots a < b. The product (x − a)(x − b) is positive outside the roots, negative between them and zero at the roots. Strict > and < exclude roots; ≤ and ≥ include them. Select the required intervals.

For c > 0, x² < c means −√c < x < √c. For x² > c, take x < −√c or x > √c. Strict inequalities exclude the boundary values.

Order the roots a < b. The product (x − a)(x − b) is positive outside the roots, negative between them and zero at the roots. Strict > and < exclude roots; ≤ and ≥ include them. Select the required intervals.

Order the roots a < b. The product (x − a)(x − b) is positive outside the roots, negative between them and zero at the roots. Strict > and < exclude roots; ≤ and ≥ include them. Select the required intervals.

Order the roots a < b. The product (x − a)(x − b) is positive outside the roots, negative between them and zero at the roots. Strict > and < exclude roots; ≤ and ≥ include them. Select the required intervals.