Linear Equations: Linear Inequalities & Bounded Regions

Q: How do you know whether an inequality line is dashed or solid?

Use a **dashed line** for strict inequalities ($ $) because boundary points are not included. Use a **solid line** for non-strict inequalities ($\le$ and $\ge$) because boundary points satisfy the inequality.

Q: How do you find the area of a triangle formed by a line and the lines $x = a, y = b$?

1. Find the right angle vertex $C(a, b)$ where $x = a$ and $y = b$ intersect. 2. Find intersection $A$ with $x = a$ and intersection $B$ with $y = b$. 3. Compute horizontal base $W = |x_B - a|$ and vertical height $H = |y_A - b|$. 4. The area is $\frac{1}{2} \cdot W \cdot H$.

Solve linear inequalities, analyze function positivity and negativity, compare linear functions, graph half-planes, and calculate areas of bounded triangle regions.

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Slope-Intercept Form & Function Notation

Guide: Linear Inequalities & Bounded Regions

Linear inequalities extend the study of linear functions from single boundary equations to inequality relationships, intervals, shaded coordinate half-planes, and geometric enclosed regions.

1. Positivity, Negativity & Comparing Two Functions

For a linear function

f(x) = ax + b

Positivity ( $f(x) > 0$ ): The interval of $x$ where the line lies above the x-axis.
Negativity ( $f(x) < 0$ ): The interval of $x$ where the line lies below the x-axis.

Comparing Two Linear Functions ( $f(x) > g(x)$ ): To find where line $f$ is above line $g$ , first find the intersection where $f(x) = g(x)$ . Then test a point or check slopes to determine the interval of $x$ where $f(x)$ is higher.

● Positive: f(x) > 0 (x > 1)● Negative: f(x) < 0 (x < 1)

The root at x = 1 separates the function into a positive interval above the x-axis (x > 1) and a negative interval below the x-axis (x < 1).

2. Algebraic Solving & Sign Flipping Rule

When solving linear inequalities like

ax + b > cx + d

Use the distributive property to expand brackets and combine like terms.
Isolate the variable terms on one side and constants on the other.

🚨 Golden Rule (Sign Flip): Whenever you multiply or divide both sides by a negative number, you MUST reverse the inequality symbol (e.g.

\le

becomes

\ge

<

becomes

>

, and vice versa).

Solid dot (●): 4 is included in the solution set (≥ or ≤)

For x ≥ 4, draw a solid dot at 4 and shade the ray extending to the right (+∞).

3. 2D Half-Plane Graphing & Shading

A linear inequality in two variables defines an entire region of the coordinate plane bounded by a line:

Boundary Line: Graph $y = mx + b$ . Draw it solid for $\le$ or $\ge$ (boundary points included), or dashed for $<$ or $>$ (boundary points excluded).

Shading the Half-Plane: Test the point $(0,0)$ (or another point not on the line). If $(0,0)$ satisfies the inequality, shade the half-plane containing $(0,0)$ ; otherwise, shade the opposite half-plane.

Test (0,0): 0 ≤ 0 + 1 (0 ≤ 1 is True ✓) → Shade side containing (0,0)

Solid boundary indicates points on the line are included (≤ or ≥).

4. Triangle Bounded by a Line and Boundary Lines ( $x = a, y = b$ )

A linear function

y = mx + c

bounded by lines

x = a

and

y = b

forms a right-angled triangle:

Right-Angle Vertex: $C(a, b)$ at the intersection of the horizontal and vertical boundary lines.
Base: Connects $C(a, b)$ to $B(x_B, b)$ along $y = b$ , with length $W = |x_B - a|$ .
Height: Connects $C(a, b)$ to $A(a, y_A)$ along $x = a$ , with length $H = |y_A - b|$ .
Area: Calculated directly by $\text{Area} = \frac{1}{2} \cdot W \cdot H$ .

📐 Triangle Area Calculation

Vertices: A(1, 3), B(3, 1), C(1, 1)

Base = |3 - 1| = 2, Height = |3 - 1| = 2, Area = (2 × 2) / 2 = 2

Learning Topics

❓ Frequently Asked Questions

What is the slope-intercept form of a linear equation?

The slope-intercept form is

y = ax + b

, where $a$ is the slope and $b$ is the y-intercept.

How do we express

y

as a function of

x

When we rewrite a linear equation to isolate

y

in the form

y = ax + b

, the value of

y

is determined entirely by

x

. This allows us to use function notation and write

y = f(x)

How do you calculate the slope of a line passing through two points?

Use the slope formula

a = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}

. Make sure to subtract the coordinates in the same order in both numerator and denominator.

When are two lines parallel?

Two lines are parallel if and only if they have the exact same slope (

a_1 = a_2

) but different y-intercepts (

b_1 \neq b_2

). Parallel lines never intersect.

How do you determine the number of solutions for a system of two lines without solving it?

Compare their slopes and y-intercepts:
- If

a_1 \neq a_2

: Exactly one solution (the lines intersect at one point).
- If

a_1 = a_2

and

b_1 \neq b_2

: No solution (the lines are parallel).
- If

a_1 = a_2

and

b_1 = b_2

: Infinitely many solutions (the equations represent the exact same line).

How do you analyze a linear function?

To analyze a linear function

y = ax + b

, follow these main steps:
1. Slope-Intercept Form: Isolate

y

to rewrite the equation as

y = ax + b

.
2. Parameters: Identify the slope

a

and the Y-intercept value

b

.
3. X- and Y-Intercepts: Calculate the y-intercept at

(0, b)

and the x-intercept at

\left(-\frac{b}{a}, 0\right)

(when

a \neq 0

).
4. Function Behavior: Determine whether the function is increasing (

a > 0

), decreasing (

a < 0

), or constant (

a = 0

).
5. Graphing: A line is completely determined by two distinct points. Since the exercise checks the required points rather than a continuous drawn line, plot the x- and y-intercepts and the required additional points that lie on the line.

How do you find the area of a triangle formed by three lines?

1. Find the 3 intersection points by solving pairwise linear equations.
2. Draw a horizontal or vertical auxiliary line through a vertex to construct a reference right-angled triangle.
3. Compute the area using Additive Method (

S_1 + S_2

) or Subtractive Method (

S_{\text{ref}} - S_1 - S_2

) depending on the geometry.

When do you flip the inequality sign in a linear inequality?

You must flip the inequality sign whenever you multiply or divide both sides by a negative number.

How do you know whether an inequality line is dashed or solid?

Use a dashed line for strict inequalities (

<

and

>

) because boundary points are not included. Use a solid line for non-strict inequalities (

\le

and

\ge

) because boundary points satisfy the inequality.

How do you find the area of a triangle formed by a line and the lines

x = a, y = b

1. Find the right angle vertex

C(a, b)

where

x = a

and

y = b

intersect.
2. Find intersection

A

with

x = a

and intersection

B

with

y = b

.
3. Compute horizontal base

W = |x_B - a|

and vertical height

H = |y_A - b|

.
4. The area is

\frac{1}{2} \cdot W \cdot H