Mastering algebra often involves grappling with a challenging concept: inequalities. Although it is intimidating, understanding inequalities is an essential skill for tackling more complex issues in maths.
In this article, we'll explain step by step what inequalities are, explore their various forms and look into techniques for solving them. This knowledge is particularly beneficial for students who are gearing up for their AQA or Edexcel Maths GCSE exams. If you require additional assistance, such as for GCSE Maths revision, consider learning with a qualified Maths GCSE tutor.
What are Inequalities?
An inequality is a mathematical expression that describes the relationship between two values that aren't identical. For a basic inequality example, consider x > 3. This inequality indicates that x can be any number greater than 3, such as 4, 5, or even a decimal like 100.03. Unlike the equation x = 3, which tells us that x is exactly 3, inequalities present a range of values that could satisfy the inequality.
Key inequality symbols in math include:
- The "less than" symbol (<), implying one number is smaller than another: 2 < 3
- The "greater than" symbol (>), indicating one number is larger than another: 5 > 2
- The "less than or equal to" symbol (≤), stating that a number could be either smaller than or equal to another number. For instance, 4x ≤ 4 means that x could be an integer value like 4, 3, 2 and so on.
- The "greater than or equal to" symbol (≥), suggesting that the number is either greater than or precisely equal to another number. An example would be x ≥ 6, where x could be 6, 7, 8, or any other number greater than 6.
Methods to solve inequalities
Inequalities can be classified into several categories:
- Linear Inequalities: Represented usually as ax + b < c or ax + b > c
- Quadratic Inequalities: Often in the form ax2 + bx + c < 0 or ax2 + bx + c > 0
- Rational Inequalities: These involve expressions like a/x > b.
- Polynomial Inequalities: A more generalized form that includes higher-degree polynomial functions
- Absolute Value Inequalities: In these, the absolute value of a variable is compared to a number, such as ∣x+3∣<5
We have summarised each inequality type below. For more in-depth information, we recommend the Wikipedia article on inequalities.
Linear Inequalities
Solving linear equations is often the easiest out of the various types of inequalities. Sometimes, you may need to solve linear inequalities by tackling multiple inequalities simultaneously. This is known as systems of inequalities. These can be graphically depicted as overlapping regions on a coordinate plane.
For instance, solving the equations x + y < 4 and x−y > 2 involves finding the region where both inequalities intersect. This technique is particularly beneficial for real-world optimization issues.
1
What is the solution set for the linear inequality 2x - 3 < 5?
Quadratic Inequalities
Quadratic inequalities may seem daunting but are often solvable through factorization or the quadratic formula. The shape of the parabola can dictate the range of values that satisfy the inequality. For example, in x2 − 4 > 0, the solution would be x < −2 or 2x > 2. Note that these solutions don't include the points -2 and 2 because the inequality symbol is strict (>).
2
What is the solution set for the quadratic inequality x2 - 4 < 0?
Rational Inequalities
Rational inequalities involve inequalities with fractions or ratios of polynomial functions. To calculate the values of a rational inequality, you first need to find the zero points. For example, look at x-2/x+3 > 0. Here, this would be x = 2 or x = -3. Next, test intervals around these points to identify the range of x values satisfying the inequality. In this case, the solution set would be 2x > 2 or 3x < −3.
3
What is the solution set for the rational inequality x/x-1 > 1?
Polynomial Inequalities
Polynomial inequalities encompass equations of higher degrees. For instance, the inequality x2 - 4x + 3 < 0 involves finding its zeros, which are x = 1 and x = 3. Test intervals around these points to find where the inequality holds true. Here, the solution would be 1 < x < 3.
4
What is the solution set for x3 - x2 < 0?
Absolute Value Inequalities
These inequalities are unique and involve expressions like ∣x − 2∣ > 5. Such inequalities imply that x must either be greater than 7 or less than -3. The absolute value creates a 'boundary,' requiring consideration of both sides when solving the inequality.
Real-World Applications of Inequalities
Inequalities have practical uses in various disciplines. Economists employ them for modelling income disparities and market trends, while engineers utilize them to solve optimization problems like minimizing waste material. Scientists often apply inequalities to outline limitations of physical systems, such as acceptable temperature ranges for specific reactions.
Frequently asked questions
Inequalities can be visually represented using a number line, which is a powerful tool for understanding the range of values that satisfy a given inequality. Here's how to go about it:
1. Draw a Number Line: First, draw a straight, horizontal line and mark it with evenly spaced numbers. These numbers serve as reference points to indicate where the inequality falls.
2. Identify the Boundary Point: The boundary point is the number that your variable is being compared to in the inequality. For instance, in x>3, the boundary point is 3.
3. Closed or Open Circle:
- Closed Circle: If your inequality includes "greater than or equal to" (≥) or "less than or equal to" (≤), then you place a filled-in (closed) circle at the boundary point on the number line. This indicates that the boundary point is included in the solution set.
Open Circle: If the inequality is "greater than" (>) or "less than" (<), then an open circle is placed at the boundary point, indicating that this point is not included in the solution set.
4. Draw the Arrow:
- Rightward Arrow: If the inequality is "greater than" or "greater than or equal to," draw an arrow pointing right, starting from your circle. This signifies that all numbers greater than the boundary point are part of the solution.
- Leftward Arrow: For "less than" or "less than or equal to," draw an arrow pointing left to indicate that all numbers less than the boundary point are in the solution set.
5. Multiple Inequalities: If you are dealing with compound inequalities, you may have arrows pointing in both directions or even overlapping segments on the number line.
6. Shading: Some people prefer shading the section of the number line that represents the solution. This makes it clear at a glance which numbers satisfy the inequality.
7. Labelling: Don't forget to label your number line with the variable in question (usually x) and any important points or intervals. This ensures that anyone reading the number line can understand what it represents.
By following these steps, you can create a clear, easy-to-understand graphical representation of any inequality on a number line.
The fundamental difference is that equations provide exact solutions, while inequalities provide a range of possible solutions.
Final Thoughts and summary
Cracking the code on simple inequalities may seem like an uphill task initially. However, with dedicated practice and expert guidance, it morphs from a hurdle into a stepping stone for your next lessons in maths. Remember, mastering inequalities is not just about solving equations; it's about understanding the realm of possible values and their limitations.
This post was updated on 19 Oct, 2023.
