Checking Boundary Values - Pre Algebra

1. What is the boundary value of the inequality $3x - 6 > 9$?

   $x = 5$. Solving the related equation $3x - 6 = 9$ gives the boundary.

2. The boundary of $x - 4 < 2$ is $x = 6$. What does substituting 6 show?

   $2 < 2$ is false, so 6 is not in the solution set and gets an open circle.

3. What does a closed circle at an endpoint on a number line show?

   The endpoint is included in the solution set. It is used with $\le$ or $\ge$.

4. What is the greatest integer that satisfies $x + 3 \le 8.5$?

   5. The solution is $x \le 5.5$, and 5 is the largest integer not above 5.5.

5. Is the point $(4, 1)$ on the boundary of $x + 2y \le 6$, and is it a solution?

   It is on the boundary and is a solution: $4 + 2(1) = 6$, and $6 \le 6$ is true.

6. A student solved $-2x \ge 8$ and got $x \ge -4$. Testing $x = 0$ gives what result, and what does it reveal?

   $0 \ge 8$, which is false, so 0 is not a solution. The symbol should have been reversed: $x \le -4$.

7. Which inequality symbols include the boundary value in the solution set?

   $\le$ and $\ge$. The boundary makes the two sides equal, and these symbols allow equality.

8. What is the least integer that satisfies $4x > 10$?

   3. The solution is $x > 2.5$, and 3 is the first integer greater than 2.5.

9. What does an open circle at an endpoint on a number line show?

   The endpoint is not included in the solution set. It is used with $<$ or $>$.

10. Substituting $x = 3$ into $2x + 1 \le 7$ gives what statement, and is 3 in the solution set?

    $7 \le 7$, which is true, so 3 is included in the solution set.

11. In the solution set $-1 < x \le 4$, which endpoints are included?

    Only 4 is included. The strict symbol $<$ excludes $-1$.

12. Why test a value on each side of a boundary value?

    To confirm the direction of the solution: the side whose test value makes the inequality true is the solution side.

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