What is the solution of 0.2 x -4 - 2x < - 0.4 and 3x + 2.7 <3 ?

x = 0.1
x < 0.2
- 2 < x < 0.1
x < - 2 and x > 0.1

hint Hint:

This is a question of compound inequalities. A compound inequality means we are given two statements. It is joined using the words “and” and “or”. We have to find the values of the variables satisfying those statements. We will simply the equations using the rules to solve inequalities.

The correct answer is: - 2 < x < 0.1

The given inequalities are as follows:
0.2x – 4 – 2x < -0.4 and 3x + 2.7 < 3
The word used to join them is “and”. So, we have to find the values of x satisfying both the statements.
When the word “and” is used, the values of the variables simultaneously satisfy both the inequalities.
When the word “or” is used, the values of the variables satisfy either of the inequalities.
We will solve the inequalities one by one.
0.2x – 4 – 2x < -0.4
We will take the variables together and solve them first.
0.2x – 2x – 4 < - 0.4
- 1.8x – 4 < - 0.4
We will isolate the variable by adding 4 to both the sides.
-1.8x – 4 + 4 < -0.4 + 4
- 1.8x < 3.6
Now, we will divide both the sides by -1.8. As we are dividing by a negative number, the inequality will flip.
$fraction numerator negative 1.8 x over denominator negative 1.8 end fraction greater than fraction numerator 3.6 over denominator negative 1.8 end fraction$
x > -2
3x + 2.7 < 3
We will isolate the variable by subtracting 2.7 from both the sides.
3x + 2.7 – 2.7 < 3 – 2.7
3x < 0.3
Dividing both the sides by 3 we get,
x < 0.1
Now, the values satisfying the two inequalities are given as follows:
x > -2
x < 0.1
We can combine the solution and write the intersection of the solution as follow:
-2 < x < 0.1
So, the solution is -2 < x < 0.1.

We have to follow the rules of inequalities to solve such questions. We have to swap the inequality when we divide it or multiply it by a negative number. In such questions, we find the values of the variables which makes the statement of inequalities true.