Question

How do you solve and write the following in interval notation: `|1 + 5x| ≤ 11`?

Answer 1

Solution: `-2.4<=x<=2` i.e `x in [-2.4,2]`

Explanation:

`| 1+5 x| <= 11 ; 1 + 5 x <= 11 or 5 x <=10 or x <= 2` OR

`| 1+5 x| <= 11 ; 1 + 5 x >= -11 or 5 x >= -12 or x >= -12/5`

or `x >= -2.4 :.` Solution: `-2.4<=x<=2` i.e `x in [-2.4,2]` [Ans]

Answer 2

The solution is `x in [-12/5, 2]`

Explanation:

The equation is

`|1+5x|<=11`

`|1+5x|-11<=0`

When `1+5x=0`

`<=>`, `5x=-1`

`<=>`, `x=-1/5`

Let's build a sign chart

`color(white)(aaaaaa)` `x` `color(white)(aaaaaaaaa)` `-oo` `color(white)(aaaaaaaaa)` `-1/5` `color(white)(aaaaaaaaaa)` `+oo`

`color(white)(aaaaaa)` `1+5x` `color(white)(aaaaaaaaaa)` `-` `color(white)(aaaaaaa)` `0` `color(white)(aaaaa)` `+`

`color(white)(aaaaaa)` `|1+5x|` `color(white)(aaaaaaa)` `-1-5x` `color(white)(aaaaaa)` ###color(white)(aaa)#`1+5x`

`color(white)(aaaaaa)` `|1+5x|-11` `color(white)(aaa)` `-12-5x` `color(white)(aaa)` ###color(white)(aaa)#`-10+5x`

Therefore,

`-12-5x<=0`, `<=>`, `5x>=-12`, `<=>`, `x<=-12/5`

`x in (-oo, -1/5)`

and

`-10+5x<=0`, `<=>`, `x<=2`

`x in (-1/5, +oo)`

The solution is

`x in [-12/5, 2]`

graph{|1+5x|-11 [-16.09, 15.95, -12.05, 3.97]}

Answer 3

`" "`

Solution to `color(red)(|1+5x|<=11` is `color(blue)(-12/5<= x <= 2`

Interval Notation: `color(green)([-12/5, 2]`

Explanation:

`" "`
`color(green)("Step 1:"`

Given the inequality: `color(red)(|1+5x|<=11`

There are two possible cases:

Case 1: `color(red)(1+5x<=11` `And`

Case 2: `color(red)(1+5x>=-11`

Case 1:

`1+5x<=11`

Subtract `color(red)1` from both sides of the inequality.

`1+5x-color(red)1<=11-color(red)1`

`cancel 1+5x-color(red)cancel 1<=11-color(red)1`

`5x<= 10`

Divide both sides of the inequality by `color(red)5`

`(5x)/color(red)5<= 10/color(red)5`

`(cancel 5x)/color(red)cancel 5<= cancel 10^color(red)2/color(red)cancel 5`

`:. x <=2 " "` Solution 1

Case 2:

`1+5x>=-11`

Subtract `color(red)1` from both sides of the inequality.

`1+5x-color(red)(1)>=-11-color(red)(1`

`cancel 1+5x-color(red)(cancel 1)>=-11-color(red)(1`

`5x>=-12`

Divide both sides of the inequality by `color(red)5`

`(5x)/color(red)5>=-(12)/color(red)5`

`(cancel 5x)/color(red)cancel 5>=-(12)/color(red)5`

`x>=-12/5" "` Solution 2

Combine both the solutions: Solution 1 and Solution 2

`color(green)( :. x <=2 and x>=-12/5" "` is our final solution.

You can also rewrite the combined solution as:

`color(blue)(-12/5<=x<=2`

In Interval Notation: `color(blue)([-12/5,2]`

`color(green)("Step 2:"`

Graph the inequality to verify our solution:

Hope it helps.