The decision and its conditions
A product can be positive when both factors are positive or when both are negative. Mark every zero and every undefined point before testing intervals. An even power may preserve the sign across a zero, so blindly alternating signs is unreliable. A denominator zero never enters a solution set, even when an inequality allows equality. For a system of inequalities, one point must satisfy every condition simultaneously. A finite list of candidate points turns that intersection requirement into a checkable decision without claiming to describe the entire feasible region.
Worked example
How many integers satisfy (x−(-3))(x−4)≤0? The roots are -3 and 4; the upward-opening product is nonpositive between them. Equality includes both endpoints, so integers run from -3 through 4. The count is 4−(-3)+1=8.
A contrasting decision
How many integers satisfy (x−2)/(x−6)≤0? The numerator is zero at 2; the denominator is zero and undefined at 6. The quotient is nonpositive for 2≤x<6. The integer solutions are 2 through 5, giving 4.
Diagnose a competing method
The first example’s tempting answer 6 follows this mistake: Use strict inequality and omit both included roots. Identify the interval from the product’s sign, then account for endpoint inclusion.
Transfer before revealing
For P(x)=(x−3)²(x−6)³(x−9)^3, at how many distinct real zeros does the graph change sign? Pause and identify the original restrictions. Solution: The roots 3, 6, and 9 have multiplicities 2, 3, and 3. Only odd multiplicity makes the polynomial change sign across a zero. There are 2 odd multiplicities, so the graph changes sign at 2 distinct zeros.
Check the boundary of the method
For a system of inequalities, one point must satisfy every condition simultaneously. A denominator zero is excluded even when the inequality includes equality. Check the solution in the original statement. A transformed equation can generate candidates rather than equivalent solutions.
How many integers satisfy (x−(-3))(x−4)≤0?
- The roots are -3 and 4; the upward-opening product is nonpositive between them.
- Equality includes both endpoints, so integers run from -3 through 4.
- The count is 4−(-3)+1=8.
Compare the other answer choices
Count only nonpositive solutions.
Use strict inequality and omit both included roots.
Count unit gaps rather than the included integer points.
Explain your thinking.
Optional written extension. Self-assessed only; it does not change your automatic practice result. These are teaching criteria, not official awarded marks.
Show substituted work, state the required domain or geometric condition, and refute a competing method. This extension is not automatically scored.
This draft stays on this page only. Use paper if you want to keep your working.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
CLEP College Algebra exam overview