Linear Inequalities

Inequality happens when we compare two numbers or two expressions and they are not the same. There are different types of inequalities, such as numeric, algebraic, or a combination of both. We call inequalities linear when they compare linear expressions. A linear expression is a mathematical expression where the variable is not multiplied by another variable or placed in the denominator, and doesn’t have any exponents like \(x^2\) or \(x^3\). We use special symbols to show how things are being compared:

≠ : not equal

＜: less than

＞: greater than

≤: less than or equal to

≥: greater than or equal to

For example, if p＜q, then p is smaller than q. If p ≤ q, then p is smaller than or equal to q. The same goes for ＞and ≥. These signs help us to understand whether the numbers are bigger, smaller, or not equal when comparing.

While solving linear inequalities, we follow some rules that are similar to solving regular equations. The rules of linear inequalities are given below: 
 

Rule 1: Adding or subtracting the same number on both sides.
Adding or subtracting the same number from both sides does not change the inequality. 

Example:
x + 5 ＜10

Subtracting 5 from both sides,
x + 5 – 5＜10 – 5
x ＜5.
 

Rule 2: Multiply and divide both sides by the same positive number
Multiplying or dividing both sides by the same positive number does not change the inequality. In x/2 ＞3, if we multiply both sides by 2 to isolate x, we get x ＞6. In 2x ＞ 10, dividing both sides by 2, we get x ＞ 5.

Rule 3: Changing sign when dividing and multiplying with negative numbers
When multiplying or dividing both sides by a negative number, we have to flip the inequality sign. 

For example, -2x < 8. Dividing both sides by -2, we get x > -4. As we see, the less-than sign will change into a greater-than sign. 

Rule 4: Place the variable on the left side

It is easier to interpret inequalities when the variable is written on the left side. So, instead of writing 5 ＞x, we can write it as x＜5. 

Follow the steps given below for solving inequalities in math: 
 

Step 1: Convert the inequality into an equation by temporarily replacing the inequality symbol with an equal sign. If the given inequality is \( x + 2 ＞5\), then the equation will become \(x + 2 = 5\).

Step 2: Solve the equation like the normal equations.

\(x + 2 = 5\)

\(x = 5 – 2\)

\(x = 3\)

Step 3: Draw a number line and mark a dot at the number that we got after solving the equation.

Step 4: If the number is not included, use an open circle (○). If the inequality includes the number, use a closed circle (●). For x ＞3, draw an open circle at 3 in the number line. 

Step 5: In the number line, if the inequality sign is greater than or greater than or equal, shade the numbers on the right side. If the inequality sign is less than or less than or equal, shade the numbers on the left. 

Step 6: Pick any number from the shaded region and substitute it into the original inequality to verify. Try x = 4 in \(x + 2 ＞5\)

\(4 + 2 ＞5\)

\(6 ＞5\)

We got that 6 is greater than 5, and satisfies the given inequality.

Step 7: The numbers that are shaded in the number line are the solutions for the given inequality.

When drawing graphs for linear inequalities, it is like drawing a line, but instead of just drawing, we have to shade the area where all the correct answers are. Given below are some of the steps of representing linear inequalities graphically:
 

Step 1: Rewrite the inequality in the form y = expression.

Step 2: Use the right line.

If the sign is ＞or ＜, draw a dashed line, which means the points on the line are not a part of the answer.

If the sign is ≥ or ≤, draw a solid line, which means the points on the line are part of the answer.

Step 3: Shade the correct side.

If the signs are ＞and ≥, shade the above line.

If the signs are ＜and ≤, shade the line below.

When we have two or more inequalities at the same time, and we need to find answers that work for all the inequalities together, it is called the system of linear inequalities. The answers are plotted in the graph that makes both inequalities true. Let’s learn about the system of inequalities with the following example:

Given: y < 4 and y > x.

Step 1:  Make both inequalities into equations.

y = 4

y = x.
 

Step 2:  Now draw the graph.

We have to draw the dashed line for both inequalities because there is no equal sign.

Then shade below the line y = 4.

Shade the above line for y = x.

Where the two shaded parts overlap, that is the solution. The points in the overlapping part are the answers that make both inequalities true at the same time.

Mastering linear inequalities becomes easier with the right techniques. These tips help you solve, simplify, and visualize inequalities accurately.

• Understand the meaning of each inequality symbol to avoid confusion between less than, greater than, and equal to conditions.
   
• Always reverse the inequality sign when multiplying or dividing both sides by a negative number to maintain the correct relationship.
   
• Represent your solution on a number line using open circles for strict inequalities (<, >) and closed circles for inclusive ones (≤, ≥).
   
• Test your solution by substituting a random value from the solution range to ensure it satisfies the given inequality.
   
• Simplify and isolate the variable on one side to make solving, comparing, and graphing inequalities easier and more accurate.

While dealing with limits or comparisons, linear inequalities are used. Here are some of the real-life applications of linear inequalities.
 

• Budgeting and finance: Inequalities help manage expenses by ensuring total spending stays within a fixed budget. 
   
• Business and profit planning: Companies use inequalities to determine minimum sales needed to achieve a target profit.
   
• Diet and nutrition: Nutritionists use inequalities to plan meals that meet calorie limits while maintaining balanced nutrition.
   
• Resource allocation: Industries apply inequalities to distribute limited resources efficiently across multiple projects.
   
• Transportation and logistics: Inequalities are used to calculate the maximum load a vehicle can carry without exceeding safety limits.