The number line below shows the solution set of an inequality: Which two inequalities represent the graph shown?
3x > -6 and x - 4 > 6 represent the graph shown.
The solution set depicted on the number line corresponds to values that satisfy both inequalities: 3x > -6 implies x > -2, and x - 4 > 6 implies x > 10. Together, these inequalities define the range of x values that result in the shaded area on the graph.
The inequality -2x > 4 simplifies to x < -2, suggesting values less than -2. The inequality 4x < -8 simplifies to x < -2 as well. Thus, this choice only represents values less than -2, which does not match the solution set shown on the number line.
This option accurately captures the solution set: 3x > -6 simplifies to x > -2, and x - 4 > 6 simplifies to x > 10. This means the correct graphing starts from -2 and extends to values greater than 10, perfectly aligning with the depicted solution set.
The inequality 4x < -8 simplifies to x < -2, indicating values less than -2, while x ≥ 6 indicates values starting from 6 and above. These disjoint ranges do not overlap and thus fail to represent the continuous solution set shown on the number line.
This choice is identical to option C and suffers from the same issues. It suggests two disjoint ranges: x < -2 and x ≥ 6. The absence of a continuous solution set fails to match the graph in question.
The correct inequalities, 3x > -6 and x - 4 > 6, effectively describe the region shaded on the number line, indicating values greater than -2 and 10, respectively. The other options either provide incorrect or disjoint intervals that do not represent the solution set accurately. Understanding how to translate inequalities into graphical representations is crucial for interpreting solution sets on number lines in mathematical contexts.
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