Overview
An inequality compares ordered quantities. Adding the same amount preserves order; multiplying or dividing by a negative number reverses order because it reflects the number line. A strict comparison excludes its boundary, while an inclusive comparison contains it.
Consider this example: Solve 3x − 4 < 8. Add 4 and divide by positive 3: 3x < 12, so x < 4. The strict sign excludes 4 because it makes the two sides equal.
The method also handles a different case: Solve −2x + 5 ≥ 11. Subtract 5 to get −2x ≥ 6. Dividing by −2 reverses the sign: x ≤ −3. Testing −4 gives 13 ≥ 11, confirming the selected side.
For a further application, consider: Assume a > 2. Solve (2 − a)x ≥ a + 1. Since a > 2, the coefficient 2 − a is negative and nonzero. Dividing reverses the comparison, giving x ≤ (a + 1)/(2 − a). The boundary is negative because its numerator is positive and denominator negative. Keeping ≥ selects the wrong half-line; changing the denominator to a − 2 also changes its sign.
Keeping the inequality direction when dividing by a negative value gives the opposite half-line. Confusing a strict sign with an inclusive sign incorrectly includes the endpoint. Solve like an equation but pause at negative multiplication or division. Test one value from the proposed solution region and one outside it in the original inequality.
A number-line choice expresses the same solution as an inequality, so check its endpoint and direction independently. For 3x − 4 < 8, adding 4 gives 3x < 12 and division by positive 3 gives x < 4. The matching picture has an open circle at 4 and a ray pointing left. A filled circle would include 4, which makes the original two sides equal and fails the strict inequality. A right-pointing ray would include values such as 5, which makes 11 < 8 false. An endpoint at 12 stops before the division step. For transfer, the solution x ≤ −3 from the negative-coefficient example in this lesson needs a filled circle at −3 and a ray to the left. Check the endpoint by substitution and one value on the ray rather than relying on the direction alone.