Ungleichungsrechner: Ungleichung lösen
Gib eine Ungleichung ein, und dieser Rechner löst sie Schritt für Schritt, schreibt die Lösung als Ungleichung und in Intervallschreibweise und zeichnet sie auf dem Zahlenstrahl. Er löst lineare, zweistufige, Doppel- und Betragsungleichungen.
Ergebnis
Gib Werte ein und tippe auf Berechnen, um das Ergebnis zu sehen.
Formel
Multiplying or dividing both sides by a negative number reverses the inequality sign.
An absolute value inequality becomes a double inequality (less than) or two separate ones joined by “or” (greater than).
So benutzt du diesen Rechner
- Type the inequality, for example 2x + 3 > 7, 5 - 3x <= 11 or |2x - 1| >= 5. Type <= for ≤ and >= for ≥.
- Press Calculate. You get the answer in inequality notation and interval notation, and a number line graph.
- On the graph, a filled dot means the end point is included (≤ or ≥) and an open dot means it isn’t (< or >).
Gelöste Beispiele
Easy: a two-step inequality, 2x + 3 > 7
Subtract 3 from both sides: 2x > 4.
Divide both sides by 2 (a positive number, so the sign stays the same): x > 2.
Check: x = 3 gives 9 > 7 ✓, and x = 1 gives 5 > 7 ✗, as expected. In interval notation the answer is (2, ∞).
Medium: dividing by a negative, 5 − 3x ≤ 11
Subtract 5 from both sides: −3x ≤ 6.
Divide by −3. Dividing by a negative number reverses the sign: x ≥ −2.
Check: x = 0 gives 5 ≤ 11 ✓, and x = −3 gives 14 ≤ 11 ✗. The answer is [−2, ∞), with a filled dot at −2.
Hard: an absolute value inequality, |2x − 1| ≥ 5
“Greater than” splits into two inequalities joined by “or”: 2x − 1 ≤ −5 or 2x − 1 ≥ 5.
Solve each: 2x ≤ −4 gives x ≤ −2, and 2x ≥ 6 gives x ≥ 3.
Check: x = 0 gives |−1| = 1, not ≥ 5 ✗; x = 4 gives |7| = 7 ≥ 5 ✓; x = −3 gives |−7| = 7 ≥ 5 ✓. The answer is (−∞, −2] ∪ [3, ∞).
Häufige Fehler
- Forgetting to reverse the sign when multiplying or dividing by a negative number: −3x ≤ 6 gives x ≥ −2, not x ≤ −2.
- Mixing up open and closed dots: < and > use an open dot; ≤ and ≥ use a filled dot.
- Solving only one case of an absolute value inequality. |2x − 1| ≥ 5 has two parts: x ≤ −2 or x ≥ 3.
- Using “and” instead of “or”: |u| < c is one interval between −c and c; |u| > c is two pieces joined by “or”.
Häufig gestellte Fragen
When do I flip the inequality sign?
Only when you multiply or divide both sides by a negative number. Adding or subtracting, or multiplying by a positive number, never changes the sign.
How do I solve absolute value inequalities?
Get the absolute value on its own. If it is less than c, write −c < inside < c and solve. If it is greater than c, solve inside < −c or inside > c. If c is negative, “less than” has no solution and “greater than” is true for every number.
What is interval notation?
A short way to write the answer: round brackets mean the end point is not included and square brackets mean it is. x ≥ −2 is [−2, ∞), and ∞ always gets a round bracket.
What do open and closed circles mean on the graph?
A closed (filled) circle means the number is part of the answer (≤ or ≥). An open circle means it isn’t (< or >). The shaded part of the line shows every other number in the answer.
Can it solve two-step and double inequalities?
Yes. Two-step inequalities like 2x + 3 > 7 and double inequalities like −1 < 2x + 1 ≤ 9 are solved step by step, doing the same thing to every part.
Does the inequality calculator graph the answer?
Yes — every answer is drawn on a number line, and given in both inequality and interval notation.