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Solving equations and inequalities

Mathematics - Grade 11 · Patterns and Algebra

Solving Equations and Inequalities

In this topic, you will learn how to solve different types of equations and inequalities. Equations are mathematical statements that show the equality of two expressions, while inequalities show the relationship between expressions that are not necessarily equal.

1. Solving Linear Equations

A linear equation is an equation of the first degree, meaning it has no exponents greater than one. The general form of a linear equation is:

ax + b = 0

where a and b are constants, and x is the variable you want to solve for.

Example 1

Consider the equation:

3x + 5 = 14

To solve for x, follow these steps:

  1. Subtract 5 from both sides:
    2. 3x + 5 - 5 = 14 - 5

    This simplifies to:

    3x = 9
  2. Next, divide both sides by 3:
    3. x = 9/3

    This gives:

    x = 3

2. Solving Linear Inequalities

A linear inequality is similar to a linear equation but uses inequality symbols (such as <, >, ≤, or ≥) instead of an equal sign. The general form is:

ax + b < c

or

ax + b > c

Example 2

Consider the inequality:

2x - 4 < 6

To solve for x, follow these steps:

  1. Add 4 to both sides:
    2. 2x - 4 + 4 < 6 + 4

    This simplifies to:

    2x < 10
  2. Now, divide both sides by 2:
    3. x < 10/2

    This gives:

    x < 5

3. Solving Quadratic Equations

A quadratic equation is an equation of the second degree and has the general form:

ax^2 + bx + c = 0

To solve quadratic equations, you can use the factorisation method, completing the square, or the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

Example 3

Consider the quadratic equation:

x^2 - 5x + 6 = 0

To solve this, we can factorise it:

Look for two numbers that multiply to 6 (the constant term) and add to -5 (the coefficient of x). These numbers are -2 and -3.

So, we can write:

(x - 2)(x - 3) = 0

Set each factor equal to zero:

  1. x - 2 = 0
    2. x = 2
  2. x - 3 = 0
    3. x = 3

The solutions are:

x = 2 or x = 3

4. Solving Quadratic Inequalities

To solve quadratic inequalities, first solve the corresponding quadratic equation. Then, determine the intervals where the inequality holds true.

Example 4

Consider the inequality:

x^2 - 5x + 6 < 0

From Example 3, we know the roots are 2 and 3. Now, we test intervals around these roots:

  • Choose a number less than 2, for example, 1:
    • 1^2 - 5(1) + 6 = 1 - 5 + 6 = 2 (positive)
  • Choose a number between 2 and 3, for example, 2.5:
    • (2.5)^2 - 5(2.5) + 6 = 6.25 - 12.5 + 6 = -0.25 (negative)
  • Choose a number greater than 3, for example, 4:
    • 4^2 - 5(4) + 6 = 16 - 20 + 6 = 2 (positive)

The inequality holds true between the roots:

2 < x < 3

5. Summary of Key Concepts

  • Linear equations can be solved using basic algebraic operations.
  • Linear inequalities require similar methods but focus on the inequality signs.
  • Quadratic equations can be solved through factorisation or the quadratic formula.
  • Quadratic inequalities involve testing intervals to find where the inequality is true.

Check your understanding

  1. Solve the equation: 4x - 7 = 9.
  2. Find the solution set for the inequality: 3x + 2 ≥ 11.
  3. Factorise and solve the quadratic equation: x^2 + 4x - 12 = 0.
  4. Determine the solution for the quadratic inequality: x^2 - 3x - 10 > 0.