Various Inequalities

A quadratic inequality just asks whether the parabola sits below or above the x-axis. The discriminant D decides whether a solution exists, lying either between the two roots or outside them. |x| < a means the points nearer the origin than a, and a system keeps only where its solutions overlap. Slide the coefficients to watch the green band grow or vanish, and read the |x| < a interval on the line.

Quadratic Inequalities — Below/Above the Parabola

The solution of ax² + bx + c ≤ 0 is the interval where the parabola is below the x-axis. First determine existence with the discriminant D.

p (x coefficient)-2

q (constant)-3

💡 Solutions

①x² + px + q ≤ 0: between the two roots (α ≤ x ≤ β)

②x² + px + q ≥ 0: outside the two roots (x ≤ α or x ≥ β)

③D < 0 with a > 0: x² + px + q > 0 always holds

Systematic Approach

Quadratic Inequality Method

f(x) = a(x − α)(x − β) ≤ 0 (a > 0 → α ≤ x ≤ β)

Factor first, then check signs

When D < 0

a > 0: always positive; a < 0: always negative

Negative discriminant means the sign doesn't flip

📐 Sign-Chart Method

①Factor f(x) into a(x − α)(x − β)

②Mark the roots α, β on the number line

③Determine the sign on each interval

④The interval(s) with the desired sign give the solution

Absolute-value Inequalities

value of a3

Absolute-value Inequality

|x| < a ⟺ -a < x < a

Distance from origin less than a

Reversed

|x| > a ⟺ x < -a or x > a

Distance from origin greater than a

Systems of Inequalities

Solutions of a System

A ∩ B (intersection of solution sets)

Solve each inequality, then take the common part

🔑 Strategy

①Solve each inequality separately

②Mark the solutions on a number line

③The overlap (intersection) is the system's solution

④No overlap → 'no solution'

Work It Out

Example 1

Solve the quadratic inequality x² − x − 6 < 0.

Factor the left side.

x² − x − 6 = (x − 3)(x + 2) < 0

An upward parabola is negative between its two roots.

−2 < x < 3

▸ −2 < x < 3

An upward quadratic is below 0 between its two roots.

Example 2

Solve the system 2x − 1 > 3 and x² − 5x + 4 ≤ 0.

Solve each inequality separately.

x > 2 and (x − 1)(x − 4) ≤ 0 ⇒ 1 ≤ x ≤ 4

Take the common range.

2 < x ≤ 4

▸ 2 < x ≤ 4

A system of inequalities asks for the intersection of the solutions.

Wrap-up

Quadratic Inequality

a(x − α)(x − β) ≤ 0, a > 0 → α ≤ x ≤ β

Parabola vs x-axis

Absolute-value Inequality

|f(x)| < a ⟺ -a < f(x) < a

Strip the absolute value to get a two-sided inequality

Grade-10 education-office assessment type

For what range of the constant k does x² − 2x + k > 0 hold for all real x?

①k < 1

②k > 1

③k ≤ 1

④k ≥ 1

⑤k > 2

▸ ② k > 1

It opens upward (leading coefficient 1 > 0), so always positive requires discriminant D < 0.

D = (−2)² − 4·1·k < 0

Solve the inequality.

4 − 4k < 0 ⇒ k > 1

🎯 Exam Points

①Quadratic: check the sign of D first

②a > 0, D < 0 → always > 0

③Absolute value: |x − a| < b → a − b < x < a + b (center a, radius b)

④System: use a number-line picture to find the intersection

⑤"For all real x" → use D ≤ 0

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Various Equations

Plane Coordinates & Lines

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