Question

if all real numbers are a solution to a compound inequality, which of the following statements about the compound inequality must be true?
a the compound inequality is an \and\ inequality with no overlap in the graphs of the individual inequality statements.

b the compound inequality is an \or\ inequality. the graphs of the individual inequalities result in a fully shaded number line.

c the compound inequality is an \or\ inequality with no overlap in the graphs of the individual inequality statements.

d the compound inequality is an \and\ inequality. the graphs of the individual inequalities result in a fully shaded number line.

Explanation:

Step1: Analyze "AND" compound inequalities

For an "AND" compound inequality, the solution is the intersection of the solutions of the individual inequalities. If there's no overlap (as in option A) or even with overlap, an "AND" inequality can't cover all real numbers unless both inequalities cover all real numbers (which is rare), so A and D are incorrect.

Step2: Analyze "OR" compound inequalities

For an "OR" compound inequality, the solution is the union of the solutions of the individual inequalities. If the graphs of the individual inequalities result in a fully shaded number line (i.e., their union is all real numbers), then all real numbers are solutions. Option B describes this correctly. Option C says there's no overlap, but even with overlap, the union can cover all real numbers, and "no overlap" isn't necessary for the union to be all real numbers.

Answer:

B. The compound inequality is an "OR" inequality. The graphs of the individual inequalities result in a fully shaded number line.