Compound inequalities: how to solve and graph

Compound inequalities contain two connected conditions, such as a temperature that must be above one value and below another. They appear in exam questions, budgeting, travel planning, scientific measurements and everyday estimates. The key is to understand how the conditions combine before deciding which points belong on the number line.

A statement using and requires both conditions to be true at the same time. A statement using or allows either condition to be true. Once that distinction is clear, solving and graphing becomes a systematic process involving algebra, interval notation and endpoint checks.

Words that decide how solutions combine

A chained inequality such as (-2 \le x < 5) means that (x) must be at least (-2) and less than (5). Its solution set is the overlap between the two conditions, so it is written as ([-2,5)). The square bracket at (-2) includes that endpoint, while the bracket at (5) excludes it.

By contrast, (x \le -2) or (x \ge 5) allows either side of the number line. Its solution set is written as ((-\infty,-2]\cup[5,\infty)). A useful sign refresher can help confirm whether a boundary point should be included.

Solving each part without changing its meaning

Treat an and statement as two separate inequalities, solve both, and then keep only values satisfying each one. For example, solve (-4 \le 2x-6 < 10). Adding (6) throughout gives (2 \le 2x < 16), and dividing the entire chain by the positive number (2) gives (1 \le x < 8).

For an or statement, solve each branch independently and keep all values from either result. If multiplication or division by a negative number is needed, reverse every inequality sign in that branch. For instance, (-3x>9) becomes (x<-3), not (x>-3). When an expression can first be shortened, the difference of squares method may reduce the amount of algebra required.

Graphing an and statement

For (1 \le x < 8), place a closed circle at (1), since (x=1) satisfies the inequality, and an open circle at (8). Shade the section between them. The result is one continuous section because the two conditions overlap.

Interval notation confirms the same result: ([1,8)). A closed circle corresponds to a square bracket, while an open circle corresponds to a round bracket. Test values such as (0), (4) and (9); only (4) belongs to the shaded section.

Graphing an or statement

Consider (x<-3) or (x\ge9). Draw an open circle at (-3) and shade to the left. Then draw a closed circle at (9) and shade to the right. Both sections remain because every value in either branch is valid.

In interval notation, this is ((-\infty,-3)\cup[9,\infty)). The symbol (\cup) represents a union. Infinity is not a boundary that can be reached, so it always appears with a round bracket rather than a square bracket.

Fractions, sign charts and excluded values

Fractions require particular care when a variable appears in the denominator. Solve (\frac{x-2}{x+4}\le1) by first recording that (x\ne-4). Moving everything to one side gives (\frac{2x-6}{x+4}\le0), with critical values at (x=-4) and (x=3).

Test the three regions separated by those values. Values below (-4) and above (3) do not satisfy the inequality, while values between them do. At (x=3), the fraction equals zero, so that endpoint is included; (x=-4) remains excluded because its denominator is zero. The solution is (-4<x\le3), or ((-4,3]).

Applying the method to Australian situations

Public transport provides a familiar setting in Sydney and Melbourne. Suppose an app shows that a traveller needs an Opal or Myki balance of at least $5 for the trip and no more than $20 to stay within a weekly budget. This becomes (5\le b\le20), giving a closed endpoint at both $5 and $20.

Weekly specials at Coles or Woolworths can also generate inequalities. A household deciding whether its grocery budget is between $40 and $60 writes (40\le g\le60). The choice of closed or open circles matters if the budget is described as needing to remain strictly below $60.

Australia’s Privacy Act 1988 is relevant when students use online homework assistance. A photographed worksheet may contain a school name, student identifier or another person’s details, so its privacy policy should be checked before upload. This legal consideration does not alter the mathematics, but it supports responsible use of digital learning tools.

A dependable checking routine

Begin by writing the logical connector beside the inequality: and means intersection, while or means union. Solve each branch, reverse signs when multiplying or dividing by a negative value, and then decide whether each boundary is included. Finally, test one value from every shaded region and one excluded endpoint.

The practical takeaway is to label the logic, preserve the inequality rules and verify the shaded regions before recording the final interval or graph.