Inequalities.
Describe a range of solutions, from a number-line interval to an unshaded graph region.
Solve linear inequalities · use endpoint and boundary conventions · identify regions in two variables
A solution can be a whole interval
An equation may name one value, while an inequality can describe a whole interval. The statement x < 3 includes every permitted number smaller than 3, not just a number immediately below it.
A number line makes the boundary visible. An open circle excludes the boundary; a filled circle includes it. The direction of the ray tells you whether larger or smaller values are allowed. Decide whether x is real or restricted to integers before listing answers.
An inequality compares quantities. On a number line, an open circle excludes an endpoint; a filled circle includes it.
Pause and explainDoes x ≤ 3 allow x = 3?
Yes. The extra line means “or equal to”, so the endpoint is included.
Solve with care when multiplying by a negative
Most balancing steps work for inequalities as they do for equations, but multiplying or dividing by a negative number reverses order. On a number line, negation reflects values across zero.
For example, 2 < 5 becomes −2 > −5 after multiplication by −1. This explains the reversed inequality sign. Keep the operation visible when solving, so you know exactly when the direction must change.
Solve 3x < 2x + 4.
Subtract 2x from both sides; the inequality direction stays the same.
Solve 5 − 2x ≥ 11.
x ≤ −3
Dividing by −2 reverses the inequality. Check x = −4: 5 − 2(−4) = 13 ≥ 11.
Solve −3 ≤ 3x − 2 < 7.
−13 ≤ x < 3
Perform the operation on all three parts of a compound inequality.
Pause and explainWhat happens to x > 4 when both sides are multiplied by −2?
It becomes −2x < −8. Multiplication by a negative reverses the ordering.
Two variables: boundaries and regions
With two variables, an inequality describes a region of points rather than an interval on a line. The corresponding equation gives the boundary separating points that satisfy it from those that do not.
Draw that boundary accurately, using a solid line when equality is allowed and a dashed line when it is excluded. Test one point away from the boundary, often the origin, to decide which side satisfies the inequality. For several inequalities, keep only the shared permitted region.
Draw the boundary as an equation. Use a broken line for < or >, and a solid line for ≤ or ≥. Test a point off the line to decide which side satisfies the inequality.
Represent x < 1 and y ≥ 1.
For x < 1, draw x = 1 dashed and shade the unwanted right side. For y ≥ 1, draw y = 1 solid and shade the unwanted side below.
The wanted region is the unshaded overlap. The point (0, 2) satisfies both; (2, 2) fails the first.
Pause and explainHow can testing (0, 0) help choose a side of a boundary?
Substitute it into the inequality. If true, its side is included; if false, the other side is included, provided the point is not on the boundary.
Your turn. Test a point.
After solving, read your answer as a set of allowed values rather than a single number. Check boundary inclusion and whether the problem requires integers.
For a graphical region, label each boundary and test a point before shading. A quick substitution of one allowed and one excluded point can reveal a reversed sign or wrong side.
Solve 7 − 3x > 16.
Reveal worked answer
List the integers satisfying −2 < 2x + 1 ≤ 7.
Reveal worked answer
A region is above the solid line y = x + 1 and to the left of the dashed line x = 2. State its inequalities.