Equations and inequalities
An equation is formed by an expression either side of an equals sign, the left hand side is equal to (i.e. has the same overall value) as the right hand side. The key to dealing with equations is to ensure that the equation remains balanced, the rule is; what you do to one side of the equality you do to the other.
In solving equations you will be expected to be able to solve both linear equations and quadratic equations with one unknown variable. You may also be asked to solve for two unknown values in this case you will need two equations which need to be solved simultaneously as shown in part (c) below.
An inequality is formed by an expression either side of an inequality sign, we have four different inequality signs; less than (<), less than or equal to (≤), greater than (>) and greater than or equal to (≥). Inequalities are dealt with in much the same way as equations, in that what you do to one side you do to the other. There is one important difference; if changing sign across an inequality or swapping sides, the direction of the inequality has to change.
(a) Solve the inequality.
(b) Mark the solution on the number line given below.
(c) John and Gemma played a new computer game called Benga. John scored two bengas minus three penalties. His total score was seven points. He made the equation 2x - 3y = 7 to represent his score. Gemma scored five bengas minus five penalties for 20 points.
(i)Make an equation to represent Gemma's score.
5x - 5y = 20 (x3)
(ii)Use simultaneous equations to find the number of points for a benga and the number of points for a penalty.
2x - 3y = 7 (x-5)
A benga is worth 5 points and a penalty is 1 point
(iii)Verify your solutions in both equations.
2x - 3y = 7
2(5) - 3(1) = 7
10 - 3 = 7
7 = 7
5x - 5y = 20
5(5) - 5(1) = 20
25 - 5 = 20
20 = 20