Equations and Inequalities
Problem — How to solve mathematical expressions containing an unknown to determine its possible values?
- Understand what an equation and an inequality are.
- Learn techniques to solve simple and compound equations and inequalities.
- Know how to interpret solutions and their sets.
- Use this knowledge to model real-world problems.
Part 1: Introduction to Equations
An equation is an equality that contains one or more unknowns. Solving an equation means finding the values of the unknowns that make this equality true.
Equations allow translating a mathematical or real-world problem into a formal language. The unknown, often denoted x, represents the value to be found.
Simple Equation Example
Consider the equation 2x + 3 = 7. We look for the value of x that satisfies this equality.
We can solve it by performing operations to isolate x :
- Subtract 3 from both sides: 2x + 3 - 3 = 7 - 3 which gives 2x = 4.
- Divide both sides by 2: x = 4 ÷ 2 so x = 2.
An equation contains an unknown to find in order to satisfy an equality. Solving it involves performing operations to isolate this unknown. Understanding this principle is essential before tackling more complex equations.
Part 2: Equation Solving Techniques
Solving an equation means transforming the equality into a simple form where the unknown is isolated, while respecting the equality property (performing the same operation on both sides).
Main steps are :
- Simplify each side by combining like terms.
- Use inverse operations (addition, subtraction, multiplication, division) to isolate the unknown.
- Be careful with special rules, such as never dividing by zero.
Example of Equation with Parentheses
Let's solve 3(x - 2) = 9 :
- Distribute: 3x - 6 = 9
- Add 6 to both sides: 3x = 15
- Divide by 3: x = 5
Equations with Fractions
To solve an equation with fractions, you can multiply both sides by the common denominator to eliminate fractions before simplifying.
Example
\frac{x}{4} + 2 = 5 :
- Multiply both sides by 4: x + 8 = 20
- Subtract 8: x = 12
Mastering operations on equations, including distribution and handling fractions, is essential for effectively solving various equations. Respecting equality at each step ensures the validity of the solutions.
Part 3: Inequalities and Solving Them
An inequality is an inequality that contains one or more unknowns. Solving an inequality means determining the set of values for the unknown that make the inequality true.
Inequalities are written with symbols like < (strictly less than), <= or ≥.
Example of an Inequality
Let's solve: 2x + 3 < 7.
- Subtract 3: 2x < 4
- Divide by 2: x < 2
The solution set includes all values strictly less than 2.
Important Note
When multiplying or dividing an inequality by a negative number, you must reverse the inequality sign. For example:
- -3x > 6.
- Dividing by -3 (a negative): x < -2 (the inequality reverses).
Inequalities add complexity due to the variable direction of the inequality sign. Understanding rules related to operations, especially reversing the inequality sign when multiplying or dividing by a negative number, is crucial. Solving an inequality yields a set of solutions, not just a single value.
Part 4: Using Solution Sets
Solutions to an equation are often a set of precise values, while for an inequality, it's an interval or a union of intervals.
A solution set refers to the value(s) the unknown can take to make the equation or inequality true.
Graphical Representation on a Number Line
For an inequality such as x < 2, the solution is shown on a number line with an open circle at 2 (not included) and an arrow to the left.
Example with Interval
For 3 < x < 5, the solution set is (3 ; 5), meaning all values strictly between 3 and 5.
Understanding how to represent and interpret solution sets is fundamental to visualize results and better grasp the scope of equations and inequalities. It also helps their application in real contexts.
Part 5: Practical Exercises and Applications
Let's practice to master equations and inequalities well.
Practical Application Example
A store sells notebooks at 2 euros each and offers a discount of 3 euros starting from 5 notebooks purchased. For how many notebooks is the total price less than 15 euros?
Let x be the number of notebooks bought. The total cost without discount (for <5 notebooks) is 2x, and with discount (for ≥5 notebooks) is 2x - 3.
Inequalities to solve:
- For x < 5: 2x < 15 so x < 7.5, but x must be an integer so x ≤ 7 and x < 5 so x ∈ \{1,2,3,4\}.
- For x ≥ 5: 2x - 3 < 15 so 2x < 18 hence x < 9. With x ≥ 5, we have x ∈ \{5,6,7,8\}.
In conclusion, the number of notebooks for a total price under 15 euros is between 1 and 8 (inclusive).
Solving equations and inequalities has practical applications, like problems from everyday life. Careful mathematical translation and analysis of solutions allow making clear and effective decisions.
This course has introduced the fundamental concepts of equations and inequalities tailored for 9th grade. Through precise definitions, rigorous methods, and progressive examples, you have gained tools to solve these mathematical expressions and interpret their solutions. Mastery of these skills is essential to approach more complex math topics and use mathematical language in real situations. Don't hesitate to practice with varied exercises to strengthen your understanding and confidence.