ACARA v9 CONTENT DESCRIPTION “solve linear inequalities and simultaneous linear equations in 2 variables; interpret solutions graphically and communicate solutions in terms of the situation”
Builds on: Expanding and Factorising: Two Directions of One Idea (AC9M10A01). Rearranging and simplifying expressions is the groundwork for solving here. This unit moves from equations with a single answer to inequalities, whose answers are ranges, and to pairs of equations solved together, whose answer is a meeting point, all of which can be pictured on a graph.
When the answer is a range, not a number
An equation such as x plus two equals five has a single answer, three. An inequality is different: it describes a whole range of values. Solving x plus two is greater than five gives x is greater than three, which is satisfied not by one number but by every value above three. On a number line this solution is drawn as a shaded ray reaching out from three, with the endpoint marked by an open circle if three itself is excluded, as in greater than, or a filled circle if it is included, as in at least. Inequalities are how mathematics describes limits and thresholds, such as a minimum height to ride or a budget you must not exceed, so reading their solutions as ranges, and picturing those ranges, is the first skill of this unit.
An inequality's answer is a range
Solving an inequality gives an interval of values, drawn as a ray on the number line with an open or filled endpoint.
The solution of an inequality is a whole range, not one number. For x at least 3, every value from 3 onward works, shown by a filled circle at 3 and a shaded ray. Switch to a strict inequality to see the open circle.
The one rule that catches everyone
Solving an inequality works much like solving an equation: you add, subtract, multiply, and divide both sides to isolate the variable. There is, however, a single rule that must never be forgotten, and it is the most common slip in the whole topic. Whenever you multiply or divide both sides by a negative number, the inequality sign reverses direction. Solving minus two x is less than six, dividing both sides by minus two, turns the less-than into a greater-than, giving x is greater than minus three, not less than. The quickest way to be sure is to test a value: x equals zero gives minus two times zero, which is zero, less than six, and zero really is greater than minus three, so the flipped sign is right. Build the habit of checking, and this trap loses its sting.
Dividing by a negative flips the sign
Multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality.
A crucial rule: when you multiply or divide an inequality by a negative number, the inequality sign must flip. Reveal the correct step to see why keeping the same sign is wrong.
Two equations, solved together
A pair of linear equations in two variables, x and y, is solved simultaneously when you find the values that make both true at once. Each equation, drawn on a grid, is a straight line, and the simultaneous solution is the single point where the two lines cross, because only there do both equations hold together. For the lines y equals x plus one and y equals five minus x, the crossing point is at x equals two and y equals three, and you can confirm it satisfies both equations. Seeing simultaneous equations as intersecting lines makes the idea concrete: usually two lines meet at exactly one point, which is why such a pair usually has exactly one solution.
Simultaneous equations: where two lines cross
The solution of two linear equations is the point that lies on both lines, where they intersect.
Two linear equations in x and y are two straight lines. Their simultaneous solution is the single point where they cross. Reveal the meeting point.
Finding the point by algebra
Reading the crossing point off a graph is quick but not always exact, so algebra gives a precise method. Substitution is the most direct: when both equations are written as y equals something, set those two expressions equal to each other, which removes y and leaves a single equation in x. From y equals x plus one and y equals five minus x, setting x plus one equal to five minus x gives two x equals four, so x equals two, and putting that back gives y equals three. The answer, the pair two and three, is exactly the point the lines share. A second method, elimination, adds or subtracts the equations to cancel a variable, and reaches the same point. Either way, algebra pins down the solution that the graph only suggests.
Solving by substitution
Substitution replaces one variable using one equation, leaving a single equation to solve, then back-substitutes.
One reliable method is substitution: since both equations give y, set the two expressions equal, solve the single equation for x, then put x back to find y. The result, x equals 2 and y equals 3, matches the crossing point exactly.
Inequalities in two variables, and communicating the answer
Inequalities, too, can involve two variables, and then their solution is not a ray but a whole region of the plane, called a half-plane. The inequality y is at least x plus one is solved by every point on or above the line y equals x plus one, the line itself being the boundary. Shading that region shows the solution at a glance, and a quick test point, such as the origin, tells you which side to shade. Finally, interpreting and communicating solutions matters as much as finding them. A solution should be stated in terms of the situation it models: not merely x is greater than three, but the team needs more than three players, or the region of prices and quantities that keeps the budget balanced. Reading a graph, solving with algebra, and explaining what the answer means together make up the work of this unit.
An inequality in two variables shades a region
A linear inequality in two variables is solved by a half-plane on one side of the boundary line.
A linear inequality in x and y is solved by a region, not a line. The line y equals x plus 1 is the boundary. Shade the half-plane to see the solution set.
Quick self-check
1. The solution of the inequality x ≥ 3 is:
2. Solve −2x < 6.
3. The simultaneous solution of two linear equations, drawn as two lines, is:
4. Using substitution, the equations y = x + 1 and y = 5 − x give:
5. The solution of the inequality y ≥ x + 1 in two variables is: