Equations and Inequalities Chapter Notes | Mathematics for Grade 10 PDF Download

Equations and Inequalities

Revision: Solving Linear Equations

Linear equations involve variables with a degree of 1, solved by isolating the variable using inverse operations. The goal is to maintain equality throughout transformations.

Key Concepts

Form: Ax + B = C or Ax + B = Cx + D, where A, B, C, D are constants, and A ≠ C.

Operations:

• Add/subtract the same value to both sides.
• Multiply/divide both sides by the same non-zero value.

Verification: Substitute the solution back to check equality.

Examples
1. x + 4 = 7:

• Subtract 4: x = 7 - 4 = 3.
• Check: 3 + 4 = 7.

2. 2x + 5 = 11:

• Subtract 5: 2x = 11 - 5 = 6.
• Divide by 2: x = 6/2 = 3.
• Check: 2 × 3 + 5 = 11.

3. 3x + 4 = 2x + 1:

• Subtract 2x: 3x - 2x = 1 - 4.
• Simplify: x = -3.
• Check: 3(-3) + 4 = -5, 2(-3) + 1 = -5.

Equations with Fractions

Example: (x/5) + 1 = 2x - 8:

• Subtract (x/5): -2x + (x/5) = -8 - 1.
• Simplify: (-9x/5) = -9.
• Multiply by -5/9: x = 5.
• Check: (5/5) + 1 = 2, 2(5) - 8 = 2.

Solving Equations with Fractions

Rational expressions involve fractions with variables in the denominator, requiring restrictions to avoid division by zero.

Process

• Restriction: Ensure the denominator ≠ 0.
• Eliminate Fractions: Multiply both sides by the denominator.
• Solve: Use linear equation methods.
• Check: Verify the solution and restriction.

Example:

(x - 3)/(3x + 1) = 2, x ∈ ℚ:

• Restriction: 3x + 1 ≠ 0, so x ≠ -1/3.
• Multiply by (3x + 1): x - 3 = 2(3x + 1).
• Expand: x - 3 = 6x + 2.
• Solve: -5x = 5, x = -1.
• Check: (-1 - 3)/(3(-1) + 1) = -4/-2 = 2, and -1 ≠ -1/3.

Solution: x = -1.

Simultaneous Equations

Simultaneous equations are pairs of linear equations with two variables (x, y), solved to find a common solution (ordered pair).

Methods

Substitution:

• Solve one equation for one variable.
• Substitute into the other equation.

Elimination:

• Adjust coefficients to eliminate one variable by addition or subtraction.
• Solve for the remaining variable, then substitute back.

Example
2x + y = 5, 3x + 2y = 4:
Substitution:

• From (1): y = 5 - 2x.
• Substitute into (2): 3x + 2(5 - 2x) = 4.
• Solve: 3x + 10 - 4x = 4, -x = -6, x = 6.
• Substitute x = 6 into y = 5 - 2x: y = 5 - 12 = -7.

Solution: (6, -7).

Elimination:

• Multiply (1) by -2: -4x - 2y = -10.
• Add to (2): (3x + 2y) + (-4x - 2y) = 4 - 10.
• Solve: -x = -6, x = 6.
• Substitute x = 6 into (2): 3(6) + 2y = 4, 2y = -14, y = -7.

Solution: (6, -7).

Linear Inequalities

Inequalities compare expressions, identifying all values that satisfy the condition, often represented graphically.

Notation

Symbols: > (greater than), < (less than), ≥ (greater than or equal to), ≤ (less than or equal to).

Set Representations:

• Number Line: Open dot (◯) for <, >; closed dot (●) for ≤, ≥.
• Interval Notation: (a, b) excludes endpoints; [a, b] includes endpoints.
• Set Builder Notation: {x | condition}, e.g., {x ∈ ℝ | x > 3}.

Solving Inequalities
Process: Similar to equations, but:

• Multiplying/dividing by a negative number reverses the inequality sign.
• Solutions are ranges, not single values.

Example: 5x - 4 > 3x + 2:

• Subtract 3x: 2x - 4 > 2.
• Add 4: 2x > 6.
• Divide by 2: x > 3.
• Check: At x = 4, 5(4) - 4 = 16, 3(4) + 2 = 14, 16 > 14.
• Graph: Open dot at x = 3, arrow right.

Examples
0 < k ≤ 2, k ∈ ℤ: Integers satisfying are {1, 2}.

{x ∈ ℝ | -3 ≤ x ≤ 5}:
Interval: [-3, 5].
Number Line: Closed dots at -3 and 5, line between.

Solving Quadratic Equations by Factorisation

Quadratic equations (ax² + bx + c = 0) are solved by factorising into linear factors and setting each to zero.

Methods

• Common Factor: Take out the highest common factor.
• Difference of Squares: a² - b² = (a + b)(a - b).
• Trinomial Factorisation: Find factors of a × c that sum to b.

Examples

1. x² - 4x + 4 = 0:

• Factorise: (x - 2)(x - 2) = 0.
• Solve: x - 2 = 0, x = 2.
• Check: 2² - 4(2) + 4 = 0.

2. x² + 3x = 0:

• Factorise: x(x + 3) = 0.
• Solve: x = 0 or x + 3 = 0, x = -3.
• Check: For x = 0, 0 + 0 = 0; for x = -3, 9 - 9 = 0.

3. x² - 121 = 0:

• Factorise: (x - 11)(x + 11) = 0.
• Solve: x = 11 or x = -11.
• Check: 11² - 121 = 0, (-11)² - 121 = 0.

Subject of the Formula

Rearranging formulae isolates a specific variable, using the same principles as solving equations.

Process

• Apply inverse operations to isolate the desired variable.
• Ensure the formula remains balanced.

Examples
v = u + at [u]: Subtract at: u = v - at.
I = V/R [V]: Multiply by R: V = I × R.
A = πr² [r]:

• Divide by π: r² = A/π.
• Take square root: r = √(A/π).

Word Problems

Word problems translate real-world scenarios into equations, solved using algebraic methods.

Strategy

• Identify Variables: Assign letters to unknowns.
• Form Equations: Translate relationships into equations.
• Solve: Use equation-solving techniques.
• Verify: Check the solution in the context.

Example
Sum of two numbers is 45, one is four times the other:

• Let one number be x, the other 4x.
• Equation: x + 4x = 45.
• Solve: 5x = 45, x = 9.
• Other number: 4 × 9 = 36.
• Check: 9 + 36 = 45.

Points to Remember

• Equations represent equality; solving finds the variable’s value.
• Linear equations (Ax + B = Cx + D) are solved by isolating x, ensuring A ≠ C.
• Fraction equations require denominator restrictions to avoid division by zero.
• Simultaneous equations find common solutions (x, y) using substitution or elimination.
• Inequalities identify value ranges, with sign reversal when multiplying/dividing by negatives.
• Quadratic equations are factorised into linear factors, where the product equals zero.
• Rearranging formulae isolates a variable using equation-solving principles.
• Word problems translate scenarios into equations, solved and verified contextually.

Difficult Words

• Equation: A statement of equality between two expressions (e.g., 2x + 3 = 7).
• Inequality: A comparison showing one expression is greater or less than another (e.g., x > 3).
• Variable: A symbol representing an unknown value (e.g., x).
• Constant: A fixed number in an equation (e.g., 5 in 2x + 5).
• Coefficient: The number multiplying a variable (e.g., 2 in 2x).
• Simultaneous Equations: Two or more equations solved together for common solutions.
• Factorisation: Writing an expression as a product of factors (e.g., x² - 4 = (x - 2)(x + 2)).
• Rational Expression: A fraction with polynomials in the numerator and denominator.
• Interval Notation: Describes a set of numbers with parentheses or brackets (e.g., [2, 7]).
• Set Builder Notation: Defines a set by its properties (e.g., {x ∈ ℝ | x > 3}).

Summary

The Equations and Inequalities chapter equips Grade 10 students with skills to solve linear and quadratic equations, simultaneous equations, and inequalities, crucial for technical applications. Students revise linear equation solving, handle fractions with denominator restrictions, and solve simultaneous equations using substitution or elimination. Inequalities are solved and represented graphically, using number lines, interval, and set builder notation. Quadratic equations are tackled via factorisation, and formulae are rearranged to isolate variables. Word problems bridge theory to real-world scenarios, enhancing problem-solving skills.