7.1 Linear Inequalities in One and Two Variables

Quick Answer: A linear inequality is solved like a linear equation, except multiplying or dividing both sides by a negative number flips the inequality symbol. On the coordinate plane, graph the boundary line first—solid for ≤/≥ and dashed for </>—then test a point (the origin is fastest) to decide which half-plane to shade. Compound inequalities combine two inequalities with 'and' (intersection, one overlapping interval) or 'or' (union, either part works).

One-Variable Linear Inequalities (MA.912.AR.2.8)

A linear inequality in one variable is a statement such as 3x − 7 < 11 or −2x + 5 ≥ 13. Solving it is identical to solving the corresponding equation—combine like terms, isolate the variable term, then divide—with one critical exception: multiplying or dividing both sides by a negative number reverses the inequality symbol.

Worked example: solve −4x + 12 > 28.

1. Subtract 12 from both sides: −4x > 16.
2. Divide both sides by −4 and flip the symbol: x < −4.

The solution set is all real numbers less than −4. On the number line, draw an open circle at −4 (because the inequality is strict, <) and shade to the left.

Number-Line Conventions

A common Florida EOC trap shows a correctly solved inequality paired with a reversed circle type. If the original inequality uses ≥, the endpoint is included, so the circle must be closed—no exceptions.

Multi-Step Inequalities with Variables on Both Sides

For 5(x − 2) + 3 ≤ 2x + 4, distribute first: 5x − 10 + 3 ≤ 2x + 4, combine like terms: 5x − 7 ≤ 2x + 4, subtract 2x from both sides: 3x − 7 ≤ 4, add 7: 3x ≤ 11, divide by 3: x ≤ 11/3. No sign flip was needed because the divisor was positive.

Two-Variable Linear Inequalities (MA.912.AR.2.7)

A two-variable linear inequality looks like y > 2x − 3 or 2x + 3y ≤ 12. Its graph is a half-plane: the set of all points (x, y) satisfying the inequality.

Step-by-Step Graphing Procedure

1. Graph the boundary line. Replace the inequality symbol with = and graph the resulting line. Use a solid line for ≤ or ≥ (points on the line are solutions) and a dashed line for < or > (points on the line are NOT solutions).
2. Pick a test point not on the line—(0, 0) is easiest when the line does not pass through the origin.
3. Substitute the test point into the original inequality. If it produces a true statement, shade the half-plane containing the test point. If it produces a false statement, shade the opposite half-plane.

Worked example: graph y ≤ −x + 4.

• Boundary: y = −x + 4, slope −1, y-intercept 4. The inequality is ≤, so the line is solid.
• Test (0, 0): 0 ≤ −0 + 4 → 0 ≤ 4, which is true. Shade the half-plane containing the origin (the region below and to the left of the line).

Boundary-Line Trap

The single most frequently missed point on Florida EOC inequality items is the boundary-line type. A strict inequality (< or >) demands a dashed boundary because points on the line do not satisfy the inequality. A non-strict inequality (≤ or ≥) demands a solid boundary. If an answer choice shows y > 3x − 2 with a solid line, eliminate it immediately.

Compound Inequalities

A compound inequality joins two inequalities with the words and or or.

'And' (Intersection)

−3 ≤ 2x − 1 < 7 is shorthand for −3 ≤ 2x − 1 and 2x − 1 < 7. Solve both parts by performing the same operation on all three segments: add 1 → −2 ≤ 2x < 8, divide by 2 → −1 ≤ x < 4. The graph is a segment from −1 (closed) to 4 (open). The solution set is the overlap of the two individual solution sets.

'Or' (Union)

x < −2 or x ≥ 5 is satisfied by any number less than −2 OR any number 5 or greater. The graph has two shaded rays pointing in opposite directions. Only one part needs to be true for the full statement to hold.

Florida EOC Compound-Inequality Traps

• A three-part inequality written as a < bx + c < d is always an 'and' statement; it never represents 'or'.
• If you split a < bx + c < d into two separate inequalities and solve each independently, you must keep the connector as 'and'—changing it to 'or' changes the meaning.
• When the coefficient of x is negative and you divide all three parts by it, every inequality symbol flips: 5 > −x > −1 becomes −5 < x < 1.

Real-World Context

Florida EOC word problems translate phrases into inequality symbols: 'at least' → ≥, 'no more than' → ≤, 'fewer than' → <, 'more than' → >. A budget constraint like 'spend no more than $30 on x notebooks at $4 each and y pens at $3 each' becomes 4x + 3y ≤ 30—a non-strict two-variable inequality with a solid boundary line.

Master the boundary-line decision and the sign-flip rule, and you have covered the two most-tested skills on this benchmark.