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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.

工作范例 · x^2 - 4 > 0

Algebra · Inequalities

Solve x^2 - 4 > 0

x^{2} - 4 > 0

一步

  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.

发送答案
\left(-\infty < x \wedge x < -2\right) \vee \left(2 < x \wedge x < \infty\right)

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如何: 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.

人们对人们提出的问题

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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