Sign and solution of equations involving an affine function

Solving f(x) = 0

For a linear function f(x) = a*x + b, where a is not equal to 0, the equation f(x) = 0 always has exactly one solution, known as the zero (or root) of f:

a*x + b = 0 -> x = -b/a

Example: for f(x) = 2x - 6, we solve 2x - 6 = 0, so 2*x = 6, hence x = 3. The zero of f is 3.

Investigating the sign of f(x)

The sign of a*x + b depends on the sign of a and the position of x relative to the zero -b/a.

x	-∞ ........ -b/a ........ +∞
if a > 0	f(x) is negative then positive
if a < 0	f(x) is positive then negative

Example: f(x) = -3*x + 9. The zero is x = 9/-(-3) = 3. As a = -3 < 0, f(x) is positive for x ≤ 3 and negative for x ≥ 3.

Comparing two linear functions

To solve f(x) = g(x) or compare f(x) and g(x), we bring everything to one side: this is again a first-degree equation or inequality.

Example: Solving 2x + 1 ≥ -x + 7 reduces to 3x ≥ 6, so x ≥ 2.

Common pitfall

When multiplying or dividing an inequality by a negative number (such as a < 0), you must reverse the direction of the inequality. Forgetting to reverse the direction is the most common mistake in this type of exercise.