Linear Inequalities and Solution Notation

Step 2: Divide by $-4$ and FLIP the inequality sign.

• $x \leq -3$

When dealing with two variables (usually $x$ and $y$), inequalities are often rearranged into the form $y = mx + c$ before graphing. The Golden Rule applies here too.

Isolate $y$ in: $3x - 2y < 6$

Step 1: Subtract $3x$ from both sides.

• $-2y < -3x + 6$

Step 2: Divide both sides by $-2$ AND flip the inequality sign.

• $y > \frac{-3x}{-2} + \frac{6}{-2}$

Step 3: Simplify the terms.

• $y > \frac{3}{2}x - 3$

Representing Inequalities on a Number Line

You can write down an endless list of numbers that fit an inequality, or you can sketch a single visual diagram that shows them all instantly. A number line provides a graphical way to represent a solution set, which is the entire range of values that make the inequality true.

The specific type of circle you draw on the number line boundary is crucial. A strict inequality ($<$ or $>$) means the boundary value itself is not included, and is represented by an open circle ($\circ$). An inclusive inequality ($\leq$ or $\geq$) means the boundary is part of the solution, represented by a closed or solid circle ($\bullet$).

Sketch the solution set for: $x > 5$

• Draw a standard number line.
• Place an open circle directly above the number $5$.
• Draw a straight line with an arrow pointing to the right to indicate the infinite direction of the set.

Sketch the solution set for: $-2 \leq x < 1$

• Place a closed (solid) circle above $-2$.
• Place an open circle above $1$.
• Connect the two circles with a single horizontal line to represent this double inequality.

Inequalities in Two Variables: Shading Regions

Why do delivery companies map out exact zones for free shipping? They define a specific feasible region by combining multiple geographical constraints. In mathematics, we represent these constraints using linear inequalities plotted on a coordinate grid.

When sketching boundary lines on a graph, use a solid line for inclusive inequalities ($\leq$, $\geq$) and a dashed or dotted line for strict inequalities ($<$, $>$). To figure out which side of the line is the "wanted" area, pick a test coordinate not on the line (like $(0, 0)$) and substitute it into the inequality. If the resulting statement is true, that side is the correct region.

Shade the region satisfying: $y \geq 2x + 1$, $x < 3$, and $y \leq 5$

Step 1: Draw the inclusive boundary $y = 2x + 1$ as a solid line.

• Substitute a test point like $(0, 5)$. Since $5 \geq 2(0) + 1$ is true, the wanted region is above the line.

Step 2: Draw the strict boundary $x = 3$ as a dashed vertical line.

• The region $x < 3$ lies to the left of this line.

Step 3: Draw the inclusive boundary $y = 5$ as a solid horizontal line.

• The region $y \leq 5$ lies below this line.

Step 4: Identify the enclosed area.

• Shade the triangular intersection of these three conditions and clearly label it as Region R.

Set Notation and Integer Solutions

Just as a computer programmer needs precise syntax to write code, mathematicians use formal set-builder notation to describe specific groups of numbers. Instead of just writing $x > 5$, formal notation packages the condition inside curly brackets.

The syntax uses curly brackets $\{ \}$ and a separator, which can be either a colon ($:$) or a vertical pipe symbol ($|$). Both are read aloud as "such that". For example, $\{x : x > 5\}$ or $\{x \mid x > 5\}$ maps the traditional result into formal syntax, meaning "the set of all $x$ such that $x$ is greater than 5".

For two-variable regions on a Cartesian plane, the element on the left side of the separator is an ordered coordinate pair $(x, y)$. Multiple constraints are listed after the separator, usually divided by commas.

Represent the set of points satisfying $x + y \leq 5$, $x > 0$, and $y > 0$ using set notation.

Step 1: Write the coordinate pair and separator.

• $\{(x, y) : \dots \}$

Step 2: Insert the constraints.

• $\{(x, y) : x + y \leq 5, x > 0, y > 0\}$

Often, exam questions will ask you to list the specific integer solutions from a given set. Remember that the universal set (all possible values being considered) might contain values that are excluded if the inequality results in an empty set (no valid values). Always pay close attention to whether the boundaries are strict or inclusive when counting your integers.