Understanding Two-Step Inequalities

Solving any inequalities follow almost similar rules to solving algebraic equations. The simplest way to solve these equations is by using inverse operations. First, you need to undo the addition or subtraction, followed by inverse operation. After that, undo the multiplication or division.

Remember that the inverse operation of subtraction is addition and vice versa. Similarly, the division is the inverse operation of multiplication and vice versa. Also, remember to reverse the inequality whenever you divide or multiply both sides by negative numbers.

Before diving into the examples, let’s check some essential formulae for two-step inequalities:

• Less than (<)
• Greater than (>) 
• Less than or equal (≤) 
• Greater than or equal (≥) 
• The not equal symbol (≠)

Inequalities help compare numbers. They are often used to determine the range of values that can satisfy the conditions of any given variable.

How To Solve Two-Step Inequalities?

Solving two-step inequalities is quite straightforward, like the single-step equations. The main aim here is to isolate the variable on one side of the inequality equation and check the solution.

Sometimes, you might have to use graphical representation to use the number line to derive the result.

Let’s take a look at an example here:

½ x + 9 < 4

Here, start with isolating x and subtract 9 from both sides of your inequality.

Or, ½ x + 9-9 < 4-9

Or, ½ x < -5

Multiply both the sides with 2,

Or, 2* ½ x < -5 * 2

Or, x < -10

Now, use a graph to plot this inequality using the number line. Here, the inequality sign is less than only. So, the point will be open here to indicate that -10 will not be part of the solution set. The following image shows the shading is on the left side of the point:

Now, you need to select a particular point within this range to test your solution. Any value that is less than the obtained result, i.e., -10, will work for your inequality equation. Let’s take -13 here.

So, the equation becomes

½ * (-13)+9 <4

Or, -6.5+9 < 4

Or, 2.5 < 4

So, this is a true inequality, and hence the result is correct. 

Example 1:

Let’s discuss how to solve 4- 1/3* t < 2. But, first, make sure to check a random number within the solution range and graph it using the number line.

Solution:

Here, follow the same steps as before.

4- 1/3* t < 2

Or, 4-1/3 * t-4 < 2-4

Or, – 1/3* t< -2

Or, – 1/3* t * -1 < -2 *-1

Or, 1/3 *t < 2

Or, 1/3 *t * 3 < 2 *3

Or, t< 6

Therefore, the value of t will be less than -6 here. With the rest of the steps, remember to check your solution while using a value from the shaded part of the graph. 

For graphing an inequality, determine if the numerical value is true for the equation and which way needs to be highlighted as the solution. If the sign is < or >, remember that the solution set doesn’t contain the value, and the open circle will indicate that. In case the ≤ or ≥, the value will be included within the solution set and have a closed or filled-in, darker circle. The following is a table highlighting these guidelines for you.

Example 2:

Solve 2x+1<7

Solution:

Let’s isolate the variable term from one side of the equation.

2x+1<7

Or, 2x +1 – 1< 7 -1 [subtracting 1 from both sides]

Or, 2x < 6 

Or x < 6/2 [dividing both sides by 2]

Or, x < 3

Therefore, the inequality is true for all the values of x, which are lesser than 3. So, you get the solution for 2x+1<7, always lesser than the value of 3.

Example 3:

Solve −3x−8≥−2

Solve:

Here, we will isolate the variable term “−3x”

−3x−8≥−2

Or, -3x-8+8≥−2+8 [adding 8 on both the sides of the inequality]

Or, -3x≥6

Or, -3x/-3≥6/-3 [dividing both sides by -3]

Or, x≤-2 [whenever you multiply/divide both sides of an inequality by any negative number, always reverse the inequality.]

So, here the solutions of the inequality −3x−8≥−2 are all the numbers that are lesser than or equal to the value of 2. 

Example 4:

Represent the solution of inequality 5x + 2y > -8 graphically.

Solution:

Here, 

5x + 2y > -8

We will be solving for x here.

Start with adding -2y on both sides of the inequality.

5x + 2y – 2y > -8 – 2y

Or, 5x > -8 – 2y

Or, x > (-2/5)y – 8/5 [dividing both sides by 5]

So, value of x is greater than (-2/5)y – 8/5.

Again, we have to solve for y.

5x + 2y > -8

Or, 5x + 2y – 5x > -8 – 5x [subtracting 5x on both the sides]

Or, 2y > -5x – 8

Or, y > (-5/2)x – 4 [dividing both the sides by 2]

So, the value of y is greater than (-5/2)x – 4

Creating the graph with these values:

Remember:

• Always isolate or separate the variables on either side of the inequality.
• If you multiply the variable by some number, divide it by the same number on the other side of the inequality.
• If you multiply or divide by any negative number, always reverse the direction of the inequality.