Question

Determine the value of k for which the inequality 5-k≤-x+k<6.3 has the solution set {x|-2.8

Answer 1

The value of k for which the inequality has the solution set {x|-2.8

Step-by-step explanation:

To determine the value of k for which the inequality 5-k≤-x+k<6.3 has the solution set -2.8<x<-0.2, we need to find the values of k that satisfy both parts of the inequality.

First, let's consider the inequality 5-k≤-x+k. Simplifying it, we get 5≤2k-x. This means that k must be less than or equal to the midpoint between 5 and x. Since the solution set is -2.8<x<-0.2, the midpoint is (5+(-0.2))/2 = 4.4/2 = 2.2. Therefore, k ≤ 2.2.

Next, let's consider the inequality -x+k<6.3. Simplifying it, we get k<x+6.3. This means that k must be less than the right endpoint of the solution set, which is -0.2. Therefore, k < -0.2.

Combining both inequalities, we have k ≤ 2.2 and k < -0.2. The values of k that satisfy both conditions are k < -0.2. Therefore, the value of k for which the inequality has the solution set x is any value less than -0.2.

Answer 2

k = 1/2

Step-by-step explanation:

The inequality 5-k≤-x+k<6.3 can be rewritten as two inequalities:

5-k ≤ -x+k

-x+k < 6.3

To solve the first inequality, we subtract k from both sides and multiply by -1, since we are dividing by a negative number:

-x ≤ -5+2k

x ≥ 5-2k

To solve the second inequality, we subtract k from both sides:

-x < 6.3-k

x > k-6.3

Now, we can combine the two inequalities to get the solution set:

k-6.3 < x ≤ 5-2k

We can see that the solution set is empty if k is greater than or equal to 5/2. Therefore, the largest possible value of k is k = (5/2)-1 = 1/2.

To check that this value of k works, we can substitute it into the original inequality:

5-1/2 ≤ -x+1/2 < 6.3

4.5 ≤ -x ≤ 5.8

-4.5 ≥ x ≥ -5.8

The solution set to this inequality is -2.8 ≤ x ≤ -4.5, which is the same as the given solution set. Therefore, the value of k for which the inequality has the solution set x is k = 1/2.