Equations and Inequalities: notes

Linear Equations

• Balance rule: do the same to both sides. 3x + 5 = 20 gives 3x = 15, so x = 5.
• Brackets first: 2(x - 3) = 10 gives x - 3 = 5, so x = 8.
• Fractions: multiply every term by the common denominator to clear them.

Simultaneous Equations

• Elimination: add or subtract the equations to remove one letter. x + y = 7 and x - y = 1 add to 2x = 8, so x = 4 and y = 3.
• Substitution: write one letter in terms of the other, then put it in the second equation.
• Check: put both answers back in the original equations.

Quadratic Equations

• Factorise: x^2 - 5x + 6 = 0 gives (x - 2)(x - 3) = 0, so x = 2 or x = 3.
• Formula: for ax^2 + bx + c = 0, x = (-b plus or minus the square root of (b^2 - 4ac)) / 2a.
• Two answers: a quadratic usually has two solutions. Give both.

Inequalities

• Same method as equations: x + 3 > 7 gives x > 4.
• Reverse the sign: multiplying or dividing by a negative flips the inequality. -2x < 6 gives x > -3.
• Number line: an open circle means the end is not included; a closed circle means it is.