Two Step Inequalities

Similar to the two step equation, two step inequalities are about solving the equation in just two steps. But unlike one step inequality, two step inequality is a bit more complicated.

In Step 1 of the problem, we use the inverse of addition or subtraction to get rid of the constant value. In Step 2 we use the inverse of multiplication or division to get the result. But remember, whatever value is added, subtracted, multiplied or divided should be done on the other side too.

But unlike two step equations, the direction of the signs needs to be considered here. For example, if you use a positive number to divide or multiply, then you can maintain the sign as is, but if you use a negative number, then you need to reverse the sign of the inequality. This is the main difference between two step equations and two step inequalities.

Example 1: Solve 2y+1<9

The first step is to eliminate the variable on the side of the equation. So, we add 1 to both sides, but the inverse is subtraction, so we subtract 1 from both sides of the equation which becomes:

2y + 1 – 1 < 9 -1
2y < 8

Now in the next step, multiply by 2 on both sides, but the inverse to multiplication is division and hence we divide by 2 on both sides:

2y / 2 < 8/2
y < 4

This literally means that the inequality stands true for y all values less than 4

Example 2: Solve -5b+2<12

The first step is to eliminate the variable on side of the equation. So, we add 2 to both sides, but the inverse is subtraction, so we subtract 2 from both sides of the equation which becomes:

-5b + 2 – 2 < 12 -2
-5b < 10

Now in the next step, multiply by 5 on both sides, but the inverse to multiplication is division and hence we divide by 2 on both sides. Remember: since we are dividing the expression by a negative number (or multiplying the expression by a negative number), the direction of the inequality sign changes.

5b / -5 > 10/-5
b > -2

This literally means that the inequality stands true for b for all values greater than -2.

Example 3 – Solve 26 < 3n + 1

Subtract 1 on both sides:

26 -1 <3n + 1 – 1
25 < 3n

divide by 3 on both sides:

25/3 < n