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.
Isibonelo esisebenza · x^2 - 4 > 0
Isigaba
- 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.
Bonisa impendulo
Manje Khetha izinkinga, noma ubhale noma udwebe zakho. Isigaba ngasinye, isithombe, impendulo ifihlakele kuze kube ubuza.
Amaphawu asetshenziswa lapha
Chofoza noma yiluphi uphawu lwesibonakaliso ukuze uthole ukuchaza okuphelele, isithombe, kanye nokuthi iyiphi inhlamvu ekhona ekhuluma ngaso.
Indlela: 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.
Imibuzo abantu bebuza
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.
Okuningi Algebra
Linear equationsQuadratic equationsSystems of equationsFactoringExpandingSimplifying expressionsFunctions and graphsExponential and logarithmic equationsPolynomial equationsAbsolute value