Question

For which values of (A) the system has at least one solution

Answer 1

The values of a that ensures that the system has at least one solution is

a > (21/127)

Step-by-step explanation

We are told to find the values of a that ensures that the system has at least one solution. The system of equations include

3 (a - 5x) < 1 + x

2 - (x/2) > 3 + 5 (x - a)

To do this, we need to solve the expressions

3 (a - 5x) < 1 + x

3a - 15x < 1 + x

We can rewrite this as

1 + x > 3a - 15x

x + 15x > 3a - 1

16x > 3a - 1

Divide both sides 16

`\begin{gathered} (16x)/(16)>(3a-1)/(16) \\ x>(3a)/(16)-(1)/(16) \end{gathered}`

We then solve the second one. But to do this, let's multiply through by 2

2 - (x/2) > 3 + 5 (x - a)

4 - x > 6 + 10 (x - a)

we can rewrite as

6 + 10 (x - a) < 4 - x

6 + 10x - 10a < 4 - x

10x + x < 10a - 6 + 4

11x < 10a - 2

Divide both sides by 11

`\begin{gathered} (11x)/(11)<(10a-2)/(11) \\ x<(10a)/(11)-(2)/(11) \end{gathered}`

The two solutions are

x > (3a/16) - (1/16)

x < (10a/11) - (2/11)

To find a lasting solution, we need the inequality sign to be the same, so, we need to multiply through one of the equations by -1

x > (3a/16) - (1/16)

-x < (-3a/16) + (1/16)

So, the system of equations become

-x < (-3a/16) + (1/16)

x < (10a/11) - (2/11)

We can then add the two equations

-x + x < (-3a/16) + (1/16) + (10a/11) - (2/11)

0 < (-3a/16) + (10a/11) + (1/16) - (2/11)

0 < -0.1875a + 0.9091a + 0.0625 - 0.1818

0 < 0.7216a - 0.1193

Rewrite

0.7216a - 0.1193 > 0

0.7216a > 0.1193

Divide both sides by 0.7216

(0.7216a/0.7216) > (0.1193/0.7216)

a > 0.1653

a > (21/127)

Hope this Helps!!!