Question

For what values of a does the following system have at least 1 solution

3(a-5x)<1+x, 2-x/2>3+5(x-a)

Answer 1

• for the first inequality:

`\dashrightarrow \: { \tt{3(a - 5x) < 1 + x}} \\ \\ { \tt{3a - 15x < 1 + x}} \\ \\ { \tt{3a < 1 + 16x}} \\ \\ { \boxed{ \tt{a = (1 + 16x)/(3) \: \: { \red{}} }}}`

• for the second inequality:

`\dashrightarrow \: { \tt{ (2 - x)/(2) > 3 + 5(x - a) }} \\`

substitute for a:

`{ \tt{ (2 - x)/(2) > 3 + 5(x - (1 + 16x)/(3) ) }} \\ \\ { \tt{ (2 - x)/(2) > 3 - (5 + 80x)/(3) }} \\ \\ { \tt{2 - x > 6 - (10 + 160x)/(3) }} \\ \\ { \tt{6 - 3x > 18 - 10 - 160x}} \\ \\ { \tt{157x > 2}} \\ \\ { \tt{x > (2)/(157) }}`

• substitute to get value of a: [ first inequality ]

`{ \tt{a < (1 + 16( (2)/(157)) )/(3) }} \\ \\{ \boxed{ \tt{a < (63)/(157) }}}`

[ second inequality ]

`{ \boxed{ \tt{a > (65)/(157) }}}`