How to Solve Compound Inequalities: And, Or and Graphing

And: find the overlap

Solve 4 < x + 2 < 8. This is shorthand for two conditions joined by “and”: 4 < x + 2 and x + 2 < 8.

Subtract 2 from all three parts: 2 < x < 6.

On a number line, put open circles at 2 and 6 and shade between them. The interval is (2, 6). For example, 4 works in both conditions. The endpoints 2 and 6 do not, because the inequalities are strict.

For a double inequality, perform the same operation on all three parts. For separate “and” conditions, solve them separately, then keep only their shared values.

Or: combine the solutions

Solve 2x + 1 ≤ -3 or 3x – 2 > 10.

1. From 2x + 1 ≤ -3, subtract 1 and divide by 2: x ≤ -2.
2. From 3x – 2 > 10, add 2 and divide by 3: x > 4.
3. Keep either region: x ≤ -2 or x > 4.

Graph a closed circle at -2 with shading to the left, and an open circle at 4 with shading to the right. The interval notation is (-∞, -2] ∪ (4, ∞). The union symbol ∪ means combine the sets. Always use parentheses with infinity.

When the “or” regions overlap

The original example 4 < x + 2 < 8 or x – 5 > 0 gives 2 < x < 6 or x > 5. These regions overlap between 5 and 6, so together they cover every value greater than 2. The simplest answer is x > 2, or (2, ∞). An “or” answer does not always have two separate pieces.

Dividing by a negative: reverse both signs

Solve -6 < -2x ≤ 8. Divide every part by -2 and reverse each comparison:

3 > x ≥ -4, which is clearer in increasing order as -4 ≤ x < 3.

The graph has a closed circle at -4, an open circle at 3, and shading between them. The interval is [-4, 3). Adding or subtracting a number does not reverse the comparison signs.

No solution or all real numbers

• x > 5 and x < 1: there is no overlap, so there is no solution. You can write ∅.
• x > 1 or x < 5: the regions together include every real number, so the solution is all real numbers, written (-∞, ∞).

Check more than one value when comparing solution sets. Test an interior value, each endpoint, and a value outside the shaded region. One working value confirms that value is a solution; it does not prove the entire set is correct.

Worksheet and practice

For printable work, choose Compound Inequalities in the free Algebra 1 worksheet collection. The linked topic PDF includes solving practice and visual number-line questions. Use this lesson to check whether each final graph should show an overlap or a union.

Review solving linear inequalities if you need the individual steps, or use the Algebra 1 learning hub for related skill review.

The existing pop-up below reviews single-inequality skills used in these compound problems.