Linear inequalities in one or two variables

Linear inequalities in one or two variables express relationships where expressions are compared using inequality symbols ($\le, \ge, <, >$), defining a range of solutions or a region on a coordinate plane rather than a single fixed value.

Linear Inequalities in One Variable

Linear inequalities in one variable contain a single unknown (e.g., $x$) raised to the first power.

• Solving: Follows the same algebraic steps as linear equations to isolate the variable.

  Example: $2x + 5 < 13 \implies 2x < 8 \implies x < 4$.

• The Golden Rule: Always flip the inequality symbol ($\le$ becomes $\ge$, $<$ becomes $>$) whenever you multiply or divide both sides by a negative number.

  Example: $-4x \ge 12 \implies x \le -3$ (Dividing by $-4$ flips $\ge$ to $\le$).

• Number Line Representation: Solutions use an open circle ($\circ$) for strict inequalities ($<, >$) or a closed circle ($\bullet$) for inclusive inequalities ($\le, \ge$), with shading showing all valid values.

  Example: $x > 2$ is graphed with an open circle at $2$ and shading extending to the right.

• Compound Inequalities: Combine two conditions into a single statement, requiring operations to be performed across all three parts simultaneously.

  Example: $-1 < 2x + 3 \le 9 \implies -4 < 2x \le 6 \implies -2 < x \le 3$.

Linear Inequalities in Two Variables

Linear inequalities in two variables involve $x$ and $y$, typically written in slope-intercept form ($y < mx + b$) or standard form ($Ax + By \le C$).

• Boundary Line: The related equation ($y = mx + b$ or $Ax + By = C$) creates a boundary line that divides the coordinate plane into two half-planes.

  Example: For $y \ge 2x - 1$, first draw the boundary line $y = 2x - 1$.

• Line Style: Use a solid line for $\le$ or $\ge$ (boundary points are included in the solution set) or a dashed line for $<$ or $>$ (boundary points are excluded).

  Example: $y > -x + 4$ uses a dashed boundary line; $y \le 3x + 2$ uses a solid boundary line.

• Shading (Half-Plane): Test a reference point—most commonly $(0, 0)$ if it does not lie on the line. If the test point yields a true statement, shade the region containing that point; if false, shade the opposite side.

  Example: Testing $(0, 0)$ in $y < 2x + 5$ gives $0 < 5$ (True), so shade the half-plane containing $(0, 0)$.

1. Always flip the inequality sign when dividing or multiplying by a negative number.

   Example: Solve $-3x < 12$. Divide by $-3$ and flip the sign: $x > -4$.

2. Choose a test point like $(0, 0)$ to quickly determine which side to shade.

   Example: Determine if $(1, 2)$ is a solution to $2x + y \le 5$. Substitute $x = 1$ and $y = 2$: $2(1) + 2 = 4 \le 5$ (True, so $(1, 2)$ is a solution).

3. Pay attention to solid versus dashed boundary lines for graph matching.

   Example: An inequality stating a maximum limit with possible equality (e.g., "at most $\$50$") uses a $\le$ symbol, which corresponds to a solid line on a graph.

4. Solve compound inequalities by performing inverse operations on all parts.

   Example: Solve $-2 \le 3x + 1 < 7$. Subtract $1$ from all parts: $-3 \le 3x < 6$. Divide all parts by $3$: $-1 \le x < 2$.

5. Interpret the shaded overlapping region (feasible region) in systems of inequalities.

   Example: A system requires $x \ge 0$, $y \ge 0$, and $x + y \le 10$. The solution is the triangular region in the first quadrant bounded by the axes and the line $x + y = 10$.