Inequalities

Inequalities describe values that are greater than, less than, at least, or at most a given number. They are important because many real situations involve ranges instead of one exact answer, such as temperature limits, budget constraints, and safe operating conditions. On number lines and coordinate planes, inequalities become visual, which helps students see all possible solutions at once. Learning to read and draw these graphs builds strong connections between algebra and geometry.

On a number line, a solution set is shown with an open or closed circle and a ray extending left or right. In two variables, an inequality like y > 2x + 1 is graphed using a boundary line and a shaded region that represents all ordered pairs that satisfy the statement. Solid boundary lines mean the line is included, while dashed lines mean it is not included. When several inequalities are combined, the overlapping shaded area is the solution region that satisfies all conditions at the same time.

Key Facts

x > a means all numbers to the right of a on a number line, using an open circle at a.
x >= a means all numbers to the right of a including a, using a closed circle at a.
x < a means all numbers to the left of a, and x <= a means all numbers to the left including a.
For linear inequalities in two variables, use the boundary line from y = mx + b, then shade above for y > or y >= and below for y < or y <=.
Use a dashed line for > or <, and a solid line for >= or <=.
If you multiply or divide an inequality by a negative number, reverse the inequality sign: if a < b, then -a > -b.

Vocabulary

Inequality: A mathematical statement that compares two quantities using symbols such as <, >, <=, or >=.
Boundary line: The line that separates points that may satisfy a two-variable inequality from points that do not.
Solution set: The collection of all values or points that make an inequality true.
Open circle: A graph marker showing that an endpoint is not included in the solution.
Solution region: The shaded part of a graph where all points satisfy the inequality or system of inequalities.

Common Mistakes to Avoid

Using a closed circle for x > 3, which is wrong because 3 is not included in the solution. Strict inequalities use an open circle or dashed boundary.
Forgetting to reverse the inequality sign after multiplying or dividing by a negative number, which gives the opposite solution. For example, from -2x > 6, the correct result is x < -3.
Shading the wrong side of a boundary line, which happens when students do not test a point. Check a simple point like (0,0) unless it lies on the line.
Drawing a solid line for y < 2x + 1, which is wrong because points on the line are not included. Use a dashed line for strict inequalities.

Practice Questions

1 Solve and graph on a number line: 3x - 5 <= 10.
2 Graph the inequality y > -2x + 4. State whether the boundary line is solid or dashed, and tell which side of the line should be shaded.
3 A system has the inequalities y >= x - 1 and y < 3. Describe in words what the overlapping solution region represents and explain whether points on each boundary are included.