Key Techniques to Solve and Graph Inequalities Efficiently
Solving linear inequalities using the graphical method is an easy way to find the solutions for linear equations. Now to solve a linear equation in one variable is easy, where we can easily plot the value in a number line. But in the case of two-variable, we need to plot the graph in an x-y plane. A linear function is involved in solving linear inequalities. A mathematical expression that contains the symbol equal-to (=) is known to be an equation. The equality symbol basically signifies that the left-hand side of the expression is equal to the right-hand side of the expression. If two mathematical expressions contain such symbols ‘<’(less than symbol), ‘>’ (greater than symbol), ‘≤’(less than or equal to symbol) or ‘≥’ (greater than or equal to symbol), they are known as inequalities. In this article we are going to discuss what is an inequality equation,solving inequalities .
Let’s say for example,
Statement 1 – Jack is 20 years old.
The equality symbol can be mathematically expressed as x= 20.
Statement 2 – Now if I say Jack’s age is at least 20 years, then this can be expressed as x ≥ 20.
Thus, Statement 1 that is given above is an equation and Statement 2 is an inequality.
Sometimes we do Need to Solve 2 Inequalities These
Solving Linear Inequalities
We have already discussed what is an inequality equation. Let’s now discuss the method of solving 2 inequalities graphically. The graph of a linear equation is basically a straight line and any point in the Cartesian plane with respect to that will lie on either side of the line. Let us consider the expression ax + by for a linear equation in two variables. The following inequalities can be framed using the expression ax+by.
ax + by ≤ c
ax + by < c
ax + by > c
ax + by ≥ c
Linear Inequalities Graphing
For solving 2 inequalities that are mentioned above, we graph the linear expression and can make the following conclusions about the inequality.
ax + by < c
The region lying below the line ax + by = c or the region that is marked as II consists of all those points that will satisfy the inequality ax + by < c. The region II is known to be the solution region for the inequality of the type ax + by < c. The line is dotted since the solution region excludes the line ax + by = c.
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ax + by ≤ c
The region that lies below and includes the line ax + by = c or the region marked as II, it consists of all those points that will satisfy the inequality ax + by ≤ c. The region II is known to be the solution region for the inequality of the type ax + by ≤ c.
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ax + by > c
The region lying in the upper half of the line ax + by = c or the region marked as I and consists of all those points that would satisfy the inequality ax + by > c. The region I is known to be the solution region for the inequality of the type ax + by > c. Since the solution region excludes the line ax + by = c, therefore we say that the line is dotted.
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ax + by ≥ c
The region lying below and including the line ax + by = c or the region marked I consist of all those points that would satisfy the inequality ax + by ≥ c. The region I is known to be the solution region for the inequality of the type ax + by ≥ c.
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Important Points you Need to Remember
You can solve simple inequalities by adding, subtracting, multiplying or dividing both sides until you are left with the variable on its own.
But these things will tend to change the direction of inequality:
When we multiply or divide both the sides by a negative number.
When we swap the right hand sides and the left hand sides.
You don't need to multiply or divide by a variable (unless you know that it is always positive or always negative).
Questions to be Solved
Question 1) Solve for the value of x and check for : x + 5 = 3
Solution)Using the same procedures learned above, we subtract 5 from each side of the equation obtaining,
x+5-5 = 3-5
Therefore, the value of x = -2
Let’s check that the value we have got is correct or not.
Putting the value of x as -2 in the equation we have,
x+5 = 3
-2+5 = 3
3 = 3
Therefore, this proves that the value of x we have got is correct since both R.H.S and L.H.S are equal.
Question 2) Solve for the value of x and check for : x + 9 = 3
Solution)Using the same procedures learned above, we subtract 9 from each side of the equation obtaining,
x + 9 - 9 = 3 - 9
x + 0 = -6
Therefore, the value of x = -6
Let’s check that the value we have got is correct or not.
Putting the value of x as -6 in the equation we have,
x + 9 = 3
-6 + 9 = 3
3 = 3
Therefore, this proves that the value of x we have got is correct since both R.H.S and L.H.S are equal.
Question 3) Solve the following inequality -2(x+3)<10
Solution) Given inequality , 2(x+3)<10
Now first we need to divide both the sides by the number 2, we get;
= -(x+3) < 5
When we open the brackets we get,
= -x-3<5
Now we need to add 3 on both the sides,
= -x-3+3 < 5+3
=-x +0 < 8
Now divide both sides by -1 to convert the inequality into a positive one.
= -x /-1 < 8 /-1
We get , x>-8.
FAQs on Solving Inequalities Made Simple: Learn Methods & Graphs
1. What is a mathematical inequality and what do the different symbols (>, <, ≥, ≤) represent?
A mathematical inequality is a statement that compares two expressions that are not equal. It uses specific symbols to show the relationship between them. The four main symbols are:
- > (Greater than): The value on the left is larger than the value on the right (e.g., x > 5).
- < (Less than): The value on the left is smaller than the value on the right (e.g., y < 10).
- ≥ (Greater than or equal to): The value on the left is larger than or equal to the value on the right.
- ≤ (Less than or equal to): The value on the left is smaller than or equal to the value on the right.
2. What is the main difference between solving an equation and solving an inequality?
The main difference lies in the nature of their solutions and a key operational rule. While an equation (like 2x = 10) typically has one or a few specific solutions (x = 5), an inequality (like 2x > 10) has an infinite range of solutions (x > 5). The critical rule difference is that when you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality symbol. This rule does not apply to solving equations.
3. What are the fundamental rules for solving a basic linear inequality?
To solve a linear inequality, you can perform the same basic operations as with equations to isolate the variable. The key rules are:
- Addition and Subtraction: You can add or subtract the same number from both sides of the inequality without changing the inequality symbol.
- Multiplication and Division (Positive Numbers): You can multiply or divide both sides by the same positive number without changing the inequality symbol.
- Multiplication and Division (Negative Numbers): If you multiply or divide both sides by a negative number, you must flip the direction of the inequality symbol (e.g., > becomes <).
4. Why must the inequality sign be reversed when multiplying or dividing by a negative number?
Reversing the inequality sign is necessary to maintain the truth of the statement. Consider the true statement 4 > 2. If you multiply both sides by -1 without reversing the sign, you get -4 > -2, which is false. Multiplying by a negative number effectively flips the numbers' positions relative to zero on the number line. What was larger becomes 'more negative' and thus smaller. Therefore, to keep the relationship between the two sides accurate, we must reverse the inequality sign to -4 < -2, which is a true statement.
5. How do you solve an inequality with variables on both sides, for example, 5x - 3 > 3x + 1?
To solve an inequality with variables on both sides, you should first gather the variable terms on one side and the constant terms on the other. For 5x - 3 > 3x + 1:
- Step 1: Subtract 3x from both sides to consolidate the variables: (5x - 3x) - 3 > 1, which simplifies to 2x - 3 > 1.
- Step 2: Add 3 to both sides to isolate the variable term: 2x > 1 + 3, which simplifies to 2x > 4.
- Step 3: Divide both sides by 2 (a positive number, so the sign stays the same): x > 2.
6. How are the solutions of linear inequalities in one variable represented on a number line?
Solutions for one-variable inequalities are shown on a number line using a circle and a shaded arrow.
- For strict inequalities (> or <), an open circle is used at the boundary point to show that the point itself is not part of the solution.
- For non-strict inequalities (≥ or ≤), a closed (filled) circle is used to show that the boundary point is included in the solution.
- An arrow is then drawn from the circle in the direction of all other possible solutions (to the right for 'greater than', to the left for 'less than').
7. What is the method for solving a system of linear inequalities in two variables graphically?
Solving a system of linear inequalities in two variables involves finding the region on a coordinate plane that satisfies all inequalities simultaneously. The steps are:
- Step 1: For each inequality, graph the corresponding linear equation (e.g., for 2x + y > 4, graph 2x + y = 4). This line is the boundary line. Use a dashed line for > or <, and a solid line for ≥ or ≤.
- Step 2: For each inequality, pick a test point (like (0,0)) not on the line to determine which side of the line to shade. Shade the half-plane that makes the inequality true.
- Step 3: The solution to the system is the overlapping shaded region, known as the feasible region. This area contains all the (x, y) coordinate pairs that satisfy every inequality in the system.
8. How do you approach solving inequalities that involve absolute values?
Solving absolute value inequalities requires splitting them into two separate cases.
- For an inequality like |ax + b| < c, you solve the compound inequality -c < ax + b < c. This means the expression inside is 'sandwiched' between -c and c.
- For an inequality like |ax + b| > c, you solve two separate inequalities: ax + b > c OR ax + b < -c. The solution is the union of these two sets.
9. What is a common mistake to avoid when solving inequalities with variables in the denominator?
A very common and critical mistake is to cross-multiply when a variable is in the denominator. You cannot do this because you don't know if the variable expression is positive or negative. Multiplying by an unknown variable could mean you should have reversed the inequality sign. The correct method is to move all terms to one side to get zero on the other, find a common denominator, and then analyse the sign of the resulting rational expression using critical points from the numerator and denominator.
10. Where are linear inequalities applied in real-world situations?
Linear inequalities are widely used to model situations involving constraints and limitations. For example:
- In business, they help determine the minimum number of units to sell to make a profit or the maximum cost for a project to stay within budget.
- In nutrition, they can be used to plan a diet that must be above a certain amount of protein but below a certain amount of fat.
- In logistics and manufacturing, they are used in linear programming to find the optimal way to allocate limited resources like time, materials, and labour to maximize profit or minimize cost.
