solve the given inequality and graph its solution on the number line
2-x/3less than or eqaul to (</) 3-2x/5
lxl less than or equal to 5
2 is less than or eqaul to lx-6l less than or equal to 5
1.
2 - x/3 < 3 - 2x/5
multiply by 15, the LCD
30 - 5x ≤ 45 - 6x
6x - 5x ≤ 45-30
x ≤ 15
2.
|x| ≤ 5
x ≤ 5 AND -x≤5
x≤5 AND x≥-5
in short notation: -5 ≤ x ≤ 5
3.
2 ≤ |x-6| ≤ 5
case1: 2 ≤ x-6≤ 5
8 ≤ x ≤ 11
case2:
2 ≤ -x+6≤ 5
-4 ≤ -x ≤ -1
4 ≥ x ≥ 1 or
1 ≤ x ≤ 4
so 8 ≤ x ≤ 11 OR 1 ≤ x ≤ 4
To solve the given inequality and graph its solution on the number line, let's break it down step by step.
1) 2 - x/3 ≤ (</) 3 - 2x/5
To solve this inequality, we can simplify it by multiplying every term by the common denominators, which in this case is 15.
Multiply each term by 15:
15(2) - 15(x/3) ≤ (</) 15(3) - 15(2x/5)
30 - 5x ≤ (</) 45 - 6x
Now we have an easier inequality to work with.
2) |x| ≤ 5
To solve this absolute value inequality, we need to consider two cases:
- Case 1: x is positive or zero: x ≥ 0. In this case, the inequality becomes x ≤ 5.
- Case 2: x is negative: x < 0. In this case, the inequality becomes -x ≤ 5.
Simplifying each case:
- Case 1: x ≤ 5
- Case 2: x ≥ -5
Now we can graph the solution on the number line:
The number line would look something like this:
<-----------(-∞)---(-5)––––––––––––(0)––(5)––—––—(+∞)----------––––>
From the number line, we can see that the solution is all values of x that are less than or equal to 5 (x ≤ 5) and all values of x that are greater than or equal to -5 (x ≥ -5). These are the solutions to the inequality.