62.5 Quadratic Inequality Solution Form

Quadratic Inequality Solution Form is the final stage of solving a quadratic inequality, taking the qualifying regions identified during sign analysis and expressing them correctly as an interval or combination of intervals, accounting for boundary inclusion and for the special cases where the solution is either every real number or no real number at all.

Strict Boundary Exclusion

Excluding Boundaries under a Strict Inequality

When the original inequality used a strict less-than or greater-than symbol, each boundary value is excluded from the final solution, represented using an open parenthesis in interval notation.

Why Exclusion Is Required Here

Since a strict inequality does not permit equality, the values that make the expression exactly zero do not satisfy the original condition and must be left out of the interval entirely.

Inclusive Boundary Inclusion

Including Boundaries under an Inclusive Inequality

When the original inequality used a less-than-or-equal-to or greater-than-or-equal-to symbol, each boundary value is included in the final solution, represented using a closed bracket in interval notation.

Why Inclusion Is Required Here

Since an inclusive inequality does permit equality, the values that make the expression exactly zero do satisfy the original condition and must be represented as part of the solution interval.

Exterior Quadratic Interval Union

Combining the Two Outer Regions

When both the leftmost and rightmost regions of the number line satisfy the inequality, the solution is written as the union of two separate, unbounded intervals.

Why a Union Is Needed

Because these two regions are separated by the middle region, which does not satisfy the inequality, they cannot be expressed as a single continuous interval and must instead be joined using the union operation.

Interior Quadratic Interval

The Region between Two Boundaries

When only the middle region of the number line satisfies the inequality, the solution is written as a single interval bounded by the two boundary values.

Correspondence to Upward-Opening Curves

This case typically corresponds to an inequality asking where an upward-opening parabola lies below the horizontal axis, since that region is confined entirely between the curve's two horizontal intercepts.

All-Real Quadratic Solution

When Every Real Number Satisfies the Inequality

If the boundary equation has no real roots, and the single sign found across the entire number line already matches the inequality's required sign, then every real number is a solution.

Why This Outcome Occurs

Because a quadratic expression with no real roots never crosses the horizontal axis, its sign is the same across its entire domain; if that single sign already satisfies the inequality, no value of the input can fail to satisfy it.

Empty Real Solution Set

When No Real Number Satisfies the Inequality

If the boundary equation has no real roots, and the single sign found across the entire number line does not match the inequality's required sign, then no real number is a solution.

Why This Outcome Occurs

Just as in the all-real case, the expression's sign is constant across its entire domain when no real roots exist; if that constant sign never matches what the inequality requires, no input value can ever satisfy the inequality.

Quadratic Inequality Interval Notation

Assembling the Final Notation

The final solution is written in interval notation, combining the correct boundary markings, the correct interval structure, and, if needed, the union of multiple intervals, to represent every value that satisfies the original inequality.

Confirming the Notation Matches the Analysis

Before finalizing, the interval notation is checked against the sign analysis and boundary marking already completed, confirming that every included region has the required sign and that every boundary is marked open or closed according to the original inequality's symbol.