Question

Fill in the blanks of the following (i) If ` -4x ge 12," then "x…-3`. (ii) If ` (-3)/4 x le -3," then "x…4`. (iii) If ` 2/(x+2) gt 0,"then " x…-2`. (iv) If ` x gt -5," then "4x…-20`. (v) If ` x gt y and z lt 0," then "-xz…-yz`. ( vi) If ` p gt 0 and q lt 0," then "p-q...p`. (vii) If ` |x+2| gt 5," then "x...-7 or x...3`. (viii) If ` -2x+1 ge 9," then "x...-4`.

Answer 1

(i) If `-4x ge 12 rArr x le -3`
(ii) If` (-3)/4 x le -3`
`rArr x ge (-3) xx 4/-3 rArr x ge 4`
(iii) If ` 2/(x+2) gt 0`

`rArr x gt - 2`
(iv) If ` x gt - 5 rArr 4x gt - 20 `
(v) If ` x gt y and z lt 0`, then
` x z lt yz ["since ",z lt 0]`
` rArr -xz gt - yz `
(vi) If ` p gt 0 and q lt 0`,
then ` p - q gt p`
e.g., consider ` 2 gt 0 and -3 lt 0`.
` rArr 2 - (-3) = 2 + 3=5 gt 2 `
(vii) If ` |x + 2| gt 5`, then
` x +2 lt - 5 or x + 2 gt 5`
`rArr x lt -5 - 2 or x gt 5 - 2 `
`rArr x lt -7 or x gt 3 `
` rArr x in (-infty,-7) cup ( 3, infty)`
(viii) If ` -2x + 1 ge 9`, then
` -2 x ge 9 -1 rArr -2x ge 8`
`rArr 2x le - 8 rArr x le -4`