Question

How do you solve `2( x - 3) - 3x \geq - 1`?

Answer 1

(`-oo, -5`]

Explanation:

`2(x-3) - 3x >= -1`
`2x-6-3x+1 >= 0`
`-x-5>=0`
`-x >= 5`
`x<=-5`
(Because if you multiply both sides of equation by -1 you must toogle the '<' by '>' and '>' by '<')
So the answer is (`-oo, -5`]

Answer 2

`x" "<=" "-5`

Explanation:

Given:`" "2(x-3)-3x" ">=" "-1`

Multiply everything inside the brackets by 2 giving:

`" "2x-6-3x" ">=" "-1`

but `2x-3x=-x` giving:

`" "color(blue)(-x-6" " >=" " -1)`

Add `color(red)(6)` to both sides

`" "color(blue)(-x color(red)(+6)-6" ">=" "color(red)(6)-1`

But `6-6=0` giving:

`" "-x+0" ">=" "5`

`" "color(green)(-x" ">=" "5)`... Equation(1)

`color(blue)("Shortcut trick")`

Multiply both sides by (-1) to make `x` positive
Note that the inequality needs to be turned round the other way.

`color(blue)(" "+x" "<=" "-5`

`" "uarr`
`" Note that the inequality is now reversed "`

Answer 3

`x<=-5`

Explanation:

Distribute the bracket , collect terms in x on the left side of the inequality and numeric values on the right side.

`rArr2x-6-3x>=-1`

`rArr-x-6>=-1`

add 6 to both sides.

`-xcancel(-6)cancel(+6)>=-1+6`

`rArr-x>=5`

To solve for x, multiply both sides by - 1

Since this is an inequation and not an equation whenever we multiply/divide by a negative value we must `color(blue)"reverse the inequality sign"`

`(-1xx-x)<=5/(-1)larrcolor(red)"reverse the sign"`

`rArrx<=-5" is the solution"`