Equations and Inequalities

Equations and Inequalities

Study Outline

Overview of Equations and Inequalities

Explanation: Introduction to solving equations and inequalities. What Will Be Taught: Techniques for solving linear, quadratic, and inequalities. Why It’s Important: Mastery of these techniques is essential for understanding more complex mathematical concepts and solving real-world problems.

Solving Linear Equations

Explanation: Understanding and solving linear equations. What Will Be Taught: Techniques such as isolation of variables and balancing equations. Why It’s Important: Linear equations are fundamental in mathematics and are used to model real-world scenarios.

Solving Quadratic Equations

Explanation: Introduction to quadratic equations and solving techniques. What Will Be Taught: Methods such as factoring, completing the square, and the quadratic formula. Why It’s Important: Quadratic equations are widely used in various fields, including physics, engineering, and economics.

Solving Inequalities

Explanation: Techniques for solving inequalities. What Will Be Taught: Methods for solving and graphing linear and quadratic inequalities. Why It’s Important: Understanding inequalities is crucial for analyzing situations where values are constrained within a range.

Study Content

Overview of Equations and Inequalities: Equations and inequalities are mathematical statements expressing equality or inequality between two expressions. They represent relationships between variables and constants, and solving them involves finding the values of the variables that satisfy the given conditions.

Solving Linear Equations: A linear equation is an equation of the first degree with no exponents greater than one. The general form of a linear equation is ax+b=0ax + b = 0ax+b=0, where aaa and bbb are constants.

1. Isolation of Variables: To solve a linear equation, isolate the variable on one side.

Example 1: Solve the equation 2x+3=72x + 3 = 72x+3=7:

2x=7−32x = 7 – 32x=7−3 2x=42x = 42x=4 x=42=2x = \frac{4}{2} = 2x=24​=2

1. Balancing Equations: Ensure both sides of the equation are balanced by performing the same operation.

Example 2: Solve the equation 5x−4=115x – 4 = 115x−4=11:

5x=11+45x = 11 + 45x=11+4 5x=155x = 155x=15 x=155=3x = \frac{15}{5} = 3x=515​=3

Solving Quadratic Equations: A quadratic equation is an equation of the second degree with an exponent of two. The general form of a quadratic equation is ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0.

1. Factoring: Factor the quadratic equation into two binomials.

Example 3: Solve the equation x2−5x+6=0x^2 – 5x + 6 = 0x2−5x+6=0 by factoring:

(x−2)(x−3)=0(x – 2)(x – 3) = 0(x−2)(x−3)=0 x=2 or x=3x = 2 \text{ or } x = 3x=2 or x=3

1. Completing the Square: Transform the equation into a perfect square trinomial.

Example 4: Solve the equation x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0 by completing the square:

x2+6x=−5x^2 + 6x = -5×2+6x=−5

Add 999 to both sides (since (62)2=9\left(\frac{6}{2}\right)^2 = 9(26​)2=9):

x2+6x+9=4x^2 + 6x + 9 = 4×2+6x+9=4 (x+3)2=4(x + 3)^2 = 4(x+3)2=4 x+3=±2x + 3 = \pm 2x+3=±2 x=−1 or x=−5x = -1 \text{ or } x = -5x=−1 or x=−5

1. Quadratic Formula: Use the quadratic formula to solve any quadratic equation.

Example 5: Solve the equation 2×2−4x−6=02x^2 – 4x – 6 = 02×2−4x−6=0 using the quadratic formula:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}x=2a−b±b2−4ac​​

Substitute a=2a = 2a=2, b=−4b = -4b=−4, and c=−6c = -6c=−6:

x=−(−4)±(−4)2−4(2)(−6)2(2)x = \frac{-(-4) \pm \sqrt{(-4)^2 – 4(2)(-6)}}{2(2)}x=2(2)−(−4)±(−4)2−4(2)(−6)​​ x=4±16+484x = \frac{4 \pm \sqrt{16 + 48}}{4}x=44±16+48​​ x=4±644x = \frac{4 \pm \sqrt{64}}{4}x=44±64​​ x=4±84x = \frac{4 \pm 8}{4}x=44±8​ x=3 or x=−1x = 3 \text{ or } x = -1x=3 or x=−1

Solving Inequalities: An inequality is a mathematical statement that compares two expressions and shows that one is greater than, less than, or not equal to the other.

1. Linear Inequalities: Solve linear inequalities using techniques similar to solving linear equations but be mindful of the direction of the inequality.

Example 6: Solve the inequality 3x−4≤53x – 4 \leq 53x−4≤5:

3x≤5+43x \leq 5 + 43x≤5+4 3x≤93x \leq 93x≤9 x≤3x \leq 3x≤3

1. Graphing Linear Inequalities: Represent the solution set on a number line or coordinate plane.

Example 7: Graph the inequality x+y≥2x + y \geq 2x+y≥2 on a coordinate plane. The solution set includes all points on and above the line x+y=2x + y = 2x+y=2.

1. Quadratic Inequalities: Solve quadratic inequalities by finding the roots and determining the intervals that satisfy the inequality.

Example 8: Solve the inequality x2−4x>5x^2 – 4x > 5×2−4x>5:

x2−4x−5>0x^2 – 4x – 5 > 0x2−4x−5>0

Factor the quadratic:

(x−5)(x+1)>0(x – 5)(x + 1) > 0(x−5)(x+1)>0

Determine the intervals where the product is positive: x<−1x < -1x<−1 or x>5x > 5x>5.

Summary: This chapter covers essential techniques for solving equations and inequalities, focusing on linear and quadratic equations. Understanding these methods is crucial for analyzing and solving mathematical problems in various contexts.