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Inequalities

Critical points, sign tests and interval notation.

An inequality is solved like an equation — find where the expression is zero — and then a sign test on each interval between those critical points decides which intervals belong to the answer. The shaded region in the graph is the solution set.

مثال عملي: x^2 - 4 > 0

Algebra · Inequalities

Solve x^2 - 4 > 0

x^{2} - 4 > 0

خطوة خطوة

  1. x^{2} - 4 > 0

    Start from the inequality.

  2. x^{2} - 4 > 0

    Move everything to the left so we compare against 0.

  3. x = -2, x = 2

    Find the critical points — where the expression equals 0 (or is undefined). They split the number line into test intervals.

  4. (-\infty,\, -2): +\;\;(-2,\, 2): -\;\;(2,\, \infty): +

    Test a point inside each interval to find the sign of the expression there.

  5. \left(-\infty, -2\right) \cup \left(2, \infty\right)

    Keep the intervals whose sign satisfies the inequality (closed endpoints for ≤ / ≥, open for < / >).

  6. \left(-\infty < x \wedge x < -2\right) \vee \left(2 < x \wedge x < \infty\right)

    The same answer written as inequalities.

كشفت الإجابة
\left(-\infty < x \wedge x < -2\right) \vee \left(2 < x \wedge x < \infty\right)

Symbols used here

\infty
infinity
Not a number: "grows without bound" in limits and intervals.
A \cup B,\ A \cap B,\ A \setminus B
union, intersection, difference
In either; in both; in A but not B.
\neg,\ \wedge,\ \vee,\ \Rightarrow,\ \Leftrightarrow
not, and, or, implies, iff
Logical connectives.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
\sup,\ \inf
supremum, infimum
Least upper bound, greatest lower bound.
\pm
plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.
\neq
not equal
The two sides are different.
\sqrt{x},\ \sqrt[n]{x}
square root, n-th root
The non-negative number whose square (n-th power) is x.
|x|
absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.
i
imaginary unit
i² = −1.
\log_b x,\ \ln x
logarithm, natural log
The exponent b must be raised to for x; ln uses base e.
\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Inequalities

  1. Start from the inequality.
  2. Move everything to the left so we compare against 0.
  3. Find the critical points — where the expression equals 0 (or is undefined). They split the number line into test intervals.
  4. Test a point inside each interval to find the sign of the expression there.
  5. Keep the intervals whose sign satisfies the inequality (closed endpoints for ≤ / ≥, open for < / >).
  6. The same answer written as inequalities.

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

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