To solve the inequality -1/2 k is greater then or equal to -5, Danny multiplied both sides by the
reciprocal of -1/2 . His solution was k is greater then or equal to 10.What mistake did he make?
Danny checked his solution, but his check did not let him know that he had
made an error.What number could he have substituted?
c. Find a number that satisfies the original inequality but not Danny’s solution
Danny forgot to reverse the inequality sign.
c. (-1/2)k >= -5.
Multiply both sides by -2:
k <= 10.
To solve the inequality -1/2 k ≥ -5, Danny chose to multiply both sides by the reciprocal of -1/2, which is -2. However, he made a mistake in the signs.
When multiplying an inequality by a negative number, the inequality sign needs to be flipped. So, in this case, when Danny multiplied both sides by -2, he should have flipped the inequality sign to get 2k ≤ 10.
To check his solution, Danny can substitute a value for k and see if the inequality holds true. Let's choose a value of k that is less than 10, such as k = 5.
Plugging this value into the original inequality, we have:
-1/2 * 5 ≥ -5
-5/2 ≥ -5
Since -5/2 is greater than -5, the inequality holds true for k = 5.
Now let's see if Danny's solution of k ≥ 10 satisfies the original inequality. Plugging in k = 10, we have:
-1/2 * 10 ≥ -5
-5 ≥ -5
Since -5 is not greater than or equal to -5, Danny's solution does not satisfy the original inequality.
To find a number that satisfies the original inequality but not Danny's solution, we can choose a value that is greater than 10. Let's try k = 15.
Plugging in k = 15 into the original inequality, we have:
-1/2 * 15 ≥ -5
-15/2 ≥ -5
Since -15/2 is not greater than or equal to -5, the number k = 15 satisfies the original inequality but not Danny's solution.
Therefore, Danny made a mistake by not flipping the inequality sign when multiplying by -2, and the number that could have been substituted for checking his solution is k = 5. Additionally, the number that satisfies the original inequality but not Danny's solution is k = 15.