Question

please anyone tell me the easyest trick that. how we can shows the inequality x ≤ -1 ? ( this for exmaple . any inequality ). on a number line

Answer 1

Explanation:

Let's say we have inequality x < a and we are represent this solution on a number line. x < a doesn't include a but all the real values of a exactly smaller than a.  We know that for any real number a, there are infinitely many solutions till negative infinity.

Now, draw a number line and mark a point a on it, (we don't know whether a is positive or not so mark only a.)

<-------------------- a ---------------->

Now the solution of our inequality x < a will be the bold region on above number line.

If, the inequality would be x ≤ a, then we also have to include a in our solution.

<-------------------- a ------------------>

Similarly we can also find the solutions for the following inequalities on number line using the number line.

• x > a
• x ≥ a

The given inequality x ≤ -1 would be represented as follow:

<-----------(-1)----------->

You can simply solve some other types of inequalities given below using the same method:

• 2x < -3
• x ≥ 4
• -4x ≤ 4

[Hint: Just make the inequality in terms of x to solve easily.]

Additional Information:

Some properties of inequality:-  

Addition or subtraction property:  

• If we have x > y then x + z > y + z

We can add or subtract any real number on both sides of inequality, sign of inequality will remain unchanged.

Multiplication or division property:  

• If we have x > y then xz > yz ( if z is +ve)
• If we have x > y then xz < yz ( if z is -ve)

Sign of inequality reverses when we multiply the both sides of inequality with a negative number and it remains same in case of positive number.

(Same goes for division also)

Reciprocal property:  

• If we have x > y then 1/x < 1/y

Sign of inequality reverses in case when we reciprocate the terms present in the inequality.