Question

When the inequalities replace the equal sign, what happens? Why? Notice the differences between all of the inequalities.

A) The solution space becomes infinite.
B) The solution space becomes a single point.
C) The solution space becomes a line.
D) The solution space becomes a region on the number line.

Answer 1

Inequalities change the solution space from a single point or line to a range of values represented as a region on the number line or in a coordinate plane.

Step-by-step explanation:

When inequalities replace the equal sign in an equation, the solution changes from being a single value or a pair of values to a range or region of values that satisfy the inequality. For example, the equation x = 3 has only one solution, whereas the inequality x > 3 has an infinite number of solutions, as any number greater than 3 would satisfy the inequality.

In the context of a number line, the solution to an inequality is typically represented by a shaded region. If we're looking at a two-variable inequality like y > 2x + 1, the solution would be represented by a region in the coordinate plane rather than just a line.

Therefore, the correct answer to the given question would be D: The solution space becomes a region on the number line or a plane, as inequalities define a range of solutions rather than a single point or a line.