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 say linear inequalities when we 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 x² or x³. 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, the inequality remains the same. 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, when we divide 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

Inequalities are easier to understand when the variable is 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: Change the inequality to an equation by 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 part and put it in the original inequality equation to see if it works. 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: Change the inequality and make it look like “y = something”.

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.

1. Make both inequalities into equations.

y = 4

y = x

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.

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

• Business and Economics: In business, linear inequalities are used for budgeting and cost control. If a company has a budget of $2000 to produce T-shirts, each shirt costs $8. The inequality of this is 8x ≤ 2000, which shows how many T-shirts (x) can be made without going over the budget.

• Health and Diet Planning: Linear inequalities are used in health and diet planning to manage calorie intake and meet nutritional goals. Dietitians use inequalities to ensure that the daily consumption of calories, sugars, and fats is below the recommended level.

• Education and Academic Planning: In education, linear inequalities are used to set minimum requirements for passing grades, attendance, and time management.