Hello, and welcome to your mathematics lesson for today. We are going to explore linear inequations, including how to represent solutions on number lines. By the end of this lesson, you will understand what inequations are, how to solve them, and how to visualize your answers.
Let us begin with the basics. An equation, as you already know, is a statement that says two things are equal. For example, x = 5 or 3x = 7. But what happens when things are not equal?
That brings us to an inequation. An inequation is a statement that tells us one quantity is not equal to another — it is either greater or smaller. We write x < 7, which we read as "x is less than 7." Or we write x > 5, read as "x is greater than 5."
Here are the four symbols you must know. The symbol < means "is less than." The symbol > means "is greater than." The symbol ≤ means "is less than or equal to." And the symbol ≥ means "is greater than or equal to."
Now, what exactly is a linear inequation? If a and b are real numbers, and a is not zero, then any statement of the form ax + b > 0, or ax + b < 0, or ax + b ≥ 0, or ax + b ≤ 0 is called a linear inequation in one variable. The word "linear" tells us that the highest power of x is one.
Before we solve inequations, we need two important ideas: the replacement set and the solution set.
The replacement set, also called the universal set, is the collection of numbers from which we are allowed to pick values for x. It could be natural numbers, whole numbers, integers, or real numbers — whatever is specified in the problem.
The solution set is different. It contains only those numbers from the replacement set that actually make the inequation true. Think of it as filtering the replacement set through the condition given.
Here is an example. Suppose we have the inequation x > 6, and our replacement set is {2, 4, 6, 8, 10}. Only 8 and 10 satisfy the condition, so the solution set is {8, 10}.
But if our replacement set changes to {1, 3, 5, 7, 9, 11}, then the solution set becomes {7, 9, 11}. Same inequation, different answer — because the replacement set matters.
Now let us learn how to solve linear inequations. There are six key properties that guide us.
First: adding the same number to both sides does not change the inequality sign. If a is greater than b, then a + c is greater than b + c. Similarly, if a is less than b, then a + c is less than b + c. This works whether c is positive or negative.
Second: subtracting the same number from both sides also leaves the inequality sign unchanged. If a exceeds b, then a - c exceeds b - c.
Third: multiplying both sides by a positive number keeps the inequality sign as it is. If a is greater than b, and c is positive, then a × c is greater than b × c.
Fourth — and this is crucial — multiplying both sides by a negative number reverses the inequality sign. If a is greater than b, but c is negative, then a × c becomes less than b × c. The sign flips from greater than to less than.
Fifth: dividing both sides by a positive number does not change the sign. If a is greater than b and c is positive, then a/c is greater than b/c. Sixth: dividing by a negative number reverses the inequality sign. If a is greater than b and c is negative, then a/c is less than b/c.
This is the property students forget most often, so pay special attention here.
Let us see these properties in action with a worked example.
Solve 12 + 6x > 0, where x is a negative integer.
We begin by isolating the term with x. Subtract 12 from both sides: 6x > −12. Now divide by 6, which is positive, so the sign stays the same: x > −2.
But remember, x must be a negative integer. The negative integers greater than −2 are just the single number −1. Therefore, the solution set is {-1}.
Here is another example with a twist. Solve 30 - 4(2x - 1) < 30, where x is a positive integer.
First, expand the brackets: 30 - 8x + 4 < 30, which simplifies to 34 - 8x < 30. Subtract 34 from both sides: −8x < −4.
Now we divide by negative 8. Because we are dividing by a negative number, we must reverse the inequality sign. So x > 1/2.
Since x must be a positive integer, the solution set is {1, 2, 3, 4, 5, ...} — all positive integers from 1 onwards.
Now we turn to representing solutions visually using number lines.
A number line is simply a straight line where we mark real numbers at equal intervals. When we solve an inequation, we can show the solution set by marking points or drawing rays on this line.
Here is how we represent different types of solutions.
If the solution includes a specific number, we use a solid dark circle at that point. This happens with "less than or equal to" or "greater than or equal to." For example, x ≤ 3 gets a solid circle at 3, with a dark line and arrow pointing left to show all numbers less than 3 are included.
If the solution excludes a specific number, we use a hollow circle. This happens with strict "less than" or "greater than." For x < 3, we put a hollow circle at 3, with the line and arrow extending left.
Let us work through one complete example with a number line.
Solve and graph -2x + 14 < 6, where x is a real number.
Subtract 14 from both sides: −2x < −8. Divide by −2, remembering to reverse the sign: x > 4.
On the number line, we place a hollow circle at 4, because 4 is not included. Then we draw a dark line with an arrow extending to the right, showing all real numbers greater than 4.
Let us recap the key takeaways from today's lesson.
First, a linear inequation involves expressions where two quantities are not equal, connected by inequality signs.
Second, the replacement set tells us where our answers can come from, while the solution set contains only the values that satisfy the inequation.
Third, when solving, adding or subtracting any number, or multiplying or dividing by a positive number, keeps the inequality sign unchanged.
Fourth, multiplying or dividing by a negative number reverses the inequality sign.
Fifth, multiplying or dividing by a negative number reverses the inequality sign — this is essential to remember. That brings us to the end of our lesson on Linear Inequations. Practice these properties carefully, especially the sign reversal rule, and you will master this topic with confidence. Keep thinking mathematically, and I look forward to seeing you in the next lesson.