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.
Töötatud näide · x^2 - 4 > 0
Samm-sammult
- x^{2} - 4 > 0
Start from the inequality.
- x^{2} - 4 > 0
Move everything to the left so we compare against 0.
- x = -2, x = 2
Find the critical points: where the expression equals 0 (or is undefined). They split the number line into test intervals.
- (-\infty,\, -2): +\;\;(-2,\, 2): -\;\;(2,\, \infty): +
Test a point inside each interval to find the sign of the expression there.
- \left(-\infty, -2\right) \cup \left(2, \infty\right)
Keep the intervals whose sign satisfies the inequality (closed endpoints for ≤ / ≥, open for < / >).
- \left(-\infty < x \wedge x < -2\right) \vee \left(2 < x \wedge x < \infty\right)
The same answer written as inequalities.
Vastuse avalikustamine
Nüüd sina. Vali probleem, või kirjuta või joonista oma. Iga samm, pilt, vastus peidetud, kuni küsid.
Tasuta konto lisab märkmeid igale õppetunnile, ülestähenduse, oma lahendatud probleemide kohta ühes kohas ja juhendaja kohta, kellelt võid küsida selle lehekülje kohta. Matemaatika ise on avatud kõigile, allkirjastatud või mitte.
Registreeru SisselogimineSiin kasutatavad sümbolid
Koputage kõiki sümboleid, mis on mõeldud täieliku definitsiooni, pildi ja iga tähe all.
Kuidas: Inequalities
- Start from the inequality.
- Move everything to the left so we compare against 0.
- Find the critical points: where the expression equals 0 (or is undefined). They split the number line into test intervals.
- Test a point inside each interval to find the sign of the expression there.
- Keep the intervals whose sign satisfies the inequality (closed endpoints for ≤ / ≥, open for < / >).
- The same answer written as inequalities.
Küsimusi, mida inimesed küsivad
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.
Rohkem Algebra
Linear equationsQuadratic equationsSystems of equationsFactoringExpandingSimplifying expressionsFunctions and graphsExponential and logarithmic equationsPolynomial equationsAbsolute value