Chapter exploration
Use the worked example below, then change one input or assumption. Predict which part of the answer must change before calculating.
Explore this related example, then return to the prediction task above.
Drag the test value. Each factor has a sign; the product's sign decides the inequality.
Choose an interval to test it. Shading marks the solution intervals.
| Factor | ||||
|---|---|---|---|---|
| x | −negative | −negative | +positive | +positive |
| x − 5 | −negative | −negative | −negative | +positive |
| x + 3 | −negative | +positive | +positive | +positive |
| product | −negative | +positive | −negative | +positive |
| Interval | x | x − 5 | x + 3 | product |
|---|---|---|---|---|
| −negative | −negative | −negative | −negative | |
| −negative | −negative | +positive | +positive | |
| +positive | −negative | +positive | −negative | |
| +positive | +positive | +positive | +positive |
(x)+(x − 5)−(x + 3)+⇒ product negative
The shaded strips are where the expression satisfies the inequality; they match the green-tinted columns of the chart.
The product changes sign at each simple root. With three simple roots, the sign alternates, and the answer is the negative intervals.