0580 · 2.6Inequalities

Inequality symbols and number lines

0:00A ticket budget does not usually tell you to spend exactly one amount. It tells you the greatest amount you can spend. Inequalities describe these limits, along with ranges of possible values. By the end of this video, you will connect inequality symbols, number lines, algebra and regions on graphs. You need to recognise positive and negative numbers, simplify expressions, and solve basic linear equations. You also need to locate points using horizontal and vertical coordinates.

0:27We will use straight lines as boundaries, without spending time reteaching how to plot them. The important new idea is deciding which values, or which points, are allowed. An inequality compares quantities without necessarily saying they are equal. Less than and greater than are strict inequalities because equality is excluded. Less than or equal to and greater than or equal to are inclusive inequalities. The extra line underneath the symbol tells you that equality is allowed.

0:53For example, x is less than four allows three, zero and negative two. It also allows values such as three point nine, but not four itself. By contrast, x is less than or equal to four includes the boundary value four. Unless a question restricts x to integers, decimal and fractional values can be solutions too. A number line makes the whole solution set visible. The inequality negative three is less than or equal to x, and x is less than one, describes everything between those boundaries.

1:23The closed circle at negative three shows that endpoint is included. The open circle at one shows that endpoint is excluded. The joining segment includes every value between the endpoints, not just the labelled whole numbers. For x greater than one, an open circle would sit at one, with a ray extending rightwards. For x less than or equal to negative three, a closed circle would sit at negative three, with a ray extending leftwards.

1:48Direction shows which values satisfy the comparison. Words often tell you which comparison to use. At most means less than or equal to, while at least means greater than or equal to. More than and fewer than describe strict limits. Translate the meaning carefully before doing any algebra, because including or excluding the boundary changes the answer.

Worked example: ticket budget

2:09Here is a worked example involving construction and interpretation. You have twenty-three pounds, entry costs five pounds, and each ride costs three pounds. Let n represent the number of rides you take. Your total cost is 5 + 3n, and that must be less than or equal to twenty-three. Subtract five from both sides to get 3n less than or equal to eighteen. Dividing both sides by three gives n less than or equal to six.

2:34Since n counts rides, it must be a whole number, zero or more. You can therefore take zero through six rides, with six as the maximum. The algebra gives a limit, but the situation tells us how to interpret it. Negative numbers of rides make no sense, even though they satisfy the final algebraic inequality. Fractions of a ride are also excluded here. Always connect the solution back to what the variable represents.

Solving linear inequalities

2:59Solving a linear inequality is usually similar to solving a linear equation. We simplify while preserving exactly the same allowed values. Adding or subtracting the same quantity on both sides preserves their order. Multiplying or dividing both sides by a positive number also preserves their order. Consider the exam-style instruction to solve 3x less than 2x + 4. Subtract 2x from both sides to collect the variable terms together. This gives 3x − 2x less than four.

3:29Simplifying the left side leaves x less than four, which describes every solution. Checking a value helps us understand that answer. Substituting three into the original inequality gives nine less than ten, which is true. At the boundary, substituting four produces twelve less than twelve, which is false. That explains why four is excluded, rather than simply reminding us to copy the original symbol. Multiplication by a negative number behaves differently because it reverses order.

3:56Two is less than six, but negative two is greater than negative six. On a number line, multiplying by negative one reflects positions across zero. The number that was further right ends up further left, so the inequality direction must reverse. Now solve 5 − 2x greater than or equal to eleven. Subtract five from both sides to obtain −2x greater than or equal to six. Divide both sides by negative two, reversing the inequality direction.

4:24The result is x less than or equal to negative three. Subtracting five did not reverse the symbol because subtraction moves both quantities equally. Dividing by negative two did reverse it because negative scaling reverses order. The rule depends on the operation applied to both sides, not simply on seeing a negative number. This distinction prevents one of the most common algebra mistakes. A compound inequality places two conditions on the same expression.

4:51Solve negative three less than or equal to 3x − 2, which is less than seven. Add two to all three parts to obtain negative one less than or equal to 3x, which is less than nine. Divide every part by three to obtain −1/3 less than or equal to x, which is less than three. The result means x must satisfy both limits at the same time. Its lower boundary is −1/3, which is included.

5:15Its upper boundary is three, which is excluded. On the number line, use a closed circle at the lower endpoint and an open circle at the upper endpoint. These algebraic methods must work without a calculator on Paper two, and they also apply on Paper four. Keep exact fractions rather than replacing them with rounded decimals. Our boundaries here are exact, so rounding would change the solution set. The usual three significant figure convention is not a reason to round an exact inequality boundary.

Graphing inequalities in two variables

5:44With two variables, a solution is a point rather than a single position on a number line. The horizontal coordinate gives x, and the vertical coordinate gives y. An equation such as y = 1 describes a boundary line. An inequality describes all the points on one side, sometimes including the boundary itself. For x less than one, draw the vertical boundary x = 1. Every point to its left has an x coordinate smaller than one, regardless of its height.

6:11Use a broken line because points on the boundary are excluded. Cambridge’s convention is to shade the unwanted region unless the question directs otherwise, so shade the right side. For y greater than or equal to one, the boundary is horizontal. Points above it have y coordinates greater than one, and points on it satisfy equality. Draw a solid line to show the boundary is included. Shade below the line because those points fail the inequality.

6:36A sloping boundary works in exactly the same way. First replace the inequality sign with an equals sign to find the boundary equation. Draw that line as solid or broken, according to whether equality is allowed. Then test a point away from the line to identify which side satisfies the inequality. To represent x + y less than or equal to six, start with the boundary x + y = 6.

7:00When x is zero, y is six, and when y is zero, x is also six. These two intercepts locate the line, which is solid because equality is included. Test the origin, where both coordinates are zero. Substitution gives 0 + 0 less than or equal to six, which is true. The side containing the origin is therefore wanted. Shade the opposite side, while leaving the boundary solid because its points also satisfy the inequality.

7:27A test point must not lie on the boundary. Every boundary point gives equality, so it cannot distinguish the two sides. The origin is convenient only when it is not on the line. Otherwise, choose another simple point and substitute its coordinates into the original inequality. The line separates the plane into two regions. Within either region, the comparison stays the same until the boundary is crossed. That is why testing one point identifies the correct side of a straight line.

7:54When y is isolated, less than usually means below and greater than means above, but substitution remains reliable.

Combined regions and inequalities

8:01An exam question might ask you to represent three inequalities on the same grid. Take x less than four, y greater than or equal to one, and x + y less than or equal to six. Draw the broken vertical boundary at x equal to four and shade to its right. The horizontal boundary at y equal to one is solid, with the region below it shaded. For the third condition, draw the solid line through the intercepts at six on each axis.

8:27Shade above that sloping line because those points have x + y greater than six. The solution is the region left unshaded by all three conditions. Label that common region clearly if the question requests it. The point with x equal to two and y equal to two lies in the common region. Its x coordinate is less than four, and its y coordinate is greater than or equal to one.

8:49Adding its coordinates gives four, which is less than or equal to six. Satisfying only one or two conditions would not be enough. A solid boundary does not automatically make every point along that entire line a solution. Those points must still satisfy the other inequalities. Similarly, a point on a broken boundary is excluded even if it passes the other tests. The graph represents all conditions together, not three separate answers.

9:14The reverse exam task is to list inequalities that define a given region. In this diagram, region R lies right of the solid boundary x = −1. It sits below the broken line y = 3, and above the solid sloping line y = x + 1. The region is to the right of a solid vertical line, so x is greater than or equal to negative one. Below the broken horizontal line, y must be less than three.

9:40Finally, being above the solid sloping line means y is greater than or equal to x + 1. Together, these three inequalities define R. Writing only the boundary equations would not answer that question. An equation identifies the dividing line, but an inequality identifies the allowed side as well. Inspect the region’s position and the boundary style separately. If the sloping line’s direction feels uncertain, test a point clearly inside the required region.

10:07Common mistakes include reversing the symbol after subtraction, forgetting to reverse it after negative division, and including strict endpoints. On graphs, students also confuse vertical boundaries with horizontal ones or shade the wanted region. Avoid these errors by checking the operation, the boundary style and a sample solution. Always follow a question’s shading instructions if they override the usual convention. An inequality describes a collection of allowed values, rather than necessarily one answer.

10:36Number lines show those values using open circles for strict boundaries and closed circles for inclusive ones. Algebra finds the limits while preserving order, except that negative multiplication or division reverses it. Graphs extend the same idea to points, using broken or solid boundaries and leaving the common wanted region unshaded.