Which two inequalities correctly represent the graph of a number line with values less than or equal to 8?

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Understand the Problem

The question asks us to identify the two inequalities that correctly represent the given number line graph. The graph shows a line extending to the left from 8, with a closed circle at 8. This indicates that x is less than or equal to 8.

Answer

A) $x \leq 8$ and $8 \geq x$
Answer for screen readers

A) $x \leq 8$ and $8 \geq x$

Steps to Solve

  1. Analyze the number line graph The number line shows a closed circle at 8 and the line extends to the left, indicating that $x$ is less than or equal to 8. This can be written as $x \leq 8$.

  2. Consider the given options We need to find another inequality equivalent to $x \leq 8$.

  3. Analyze option A: $x \leq 8$ and $8 \geq x$ $x \leq 8$ means x is less than or equal to 8. $8 \geq x$ means 8 is greater than or equal to x, which is the same as x is less than or equal to 8. Therefore, $x \leq 8$ and $8 \geq x$ are equivalent.

  4. Analyze option B: $x < 8$ and $8 > x$ $x < 8$ means x is less than 8. This does NOT include 8. $8 > x$ means 8 is greater than x, which is the same as x is less than 8. Note: The number line includes 8. So this option is incorrect.

  5. Analyze option C: $x < 8$ and $8 < x$ $x < 8$ means x is less than 8. $8 < x$ means 8 is less than x, which is the same as x is greater than 8. These two statements are contradictory and do not represent the graph. So this option is incorrect.

  6. Analyze option D: $x \leq 8$ and $8 \leq x$ $x \leq 8$ means x is less than or equal to 8. $8 \leq x$ means 8 is less than or equal to x, which is the same as x is greater than or equal to 8. These two statements are contradictory (except for the case where x=8) and do not accurately represent the graph for values less than 8. So this option is incorrect.

  7. Final Answer Option A is the only correct option.

A) $x \leq 8$ and $8 \geq x$

More Information

The two inequalities $x \leq 8$ and $8 \geq x$ are equivalent ways of expressing that $x$ is less than or equal to 8.

Tips

A common mistake is to confuse "less than" with "less than or equal to", and to misinterpret the direction of the inequality sign. For example, $x < 8$ means x can be any number smaller than 8, but not 8 itself. $x \leq 8$ includes 8.

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