Graphing y > x - 2

Graph the inequality $y > x - 2$ on a coordinate plane.

Identify the boundary line: Treat it as $y = x - 2$. Slope $m = 1$, y-intercept $b = -2$ = Boundary: $y = x - 2$

Determine the line style: Since the symbol is $>$ (not $\geq$), points ON the line are NOT included = Use a DASHED line

Draw the boundary line: Plot y-intercept $(0, -2)$, use slope to find $(1, -1)$, $(2, 0)$ = Dashed line through these points

Choose a test point: Use $(0, 0)$ since it's easy to calculate and not on the line = Test point: $(0, 0)$

Substitute into inequality: $0 > 0 - 2 \Rightarrow 0 > -2$ TRUE! = $(0, 0)$ is a solution

Shade the correct region: Since $(0, 0)$ works, shade the side containing $(0, 0)$ = Shade ABOVE the line

Answer: Dashed line through $(0, -2)$ with slope 1, shaded above the line.

Graphing 2x + y ≤ 4

Graph the inequality $2x + y \leq 4$ on a coordinate plane.

Rewrite in slope-intercept form: $y \leq -2x + 4$ = Slope $m = -2$, y-intercept $b = 4$

Determine the line style: Since the symbol is $\leq$, points ON the line ARE included = Use a SOLID line

Draw the boundary line: Plot $(0, 4)$, use slope $-2$ to find $(1, 2)$, $(2, 0)$ = Solid line through these points

Test point $(0, 0)$: $2(0) + 0 \leq 4 \Rightarrow 0 \leq 4$ TRUE! = $(0, 0)$ is a solution

Shade the correct region: Since $(0, 0)$ works, shade the side containing the origin = Shade BELOW the line

Answer: Solid line through $(0, 4)$ and $(2, 0)$, shaded below (including the line).

Graphing y < -½x + 3

Graph the inequality $y < -\frac{1}{2}x + 3$.

Identify slope and y-intercept: Slope $m = -\frac{1}{2}$, y-intercept $b = 3$ = Boundary: $y = -\frac{1}{2}x + 3$

Determine line style: Symbol is $<$ (strict inequality), so points on line are NOT solutions = Use a DASHED line

Plot the boundary line: Start at $(0, 3)$. Go right 2, down 1 to get $(2, 2)$. Continue to $(4, 1)$ = Dashed line drawn

Test point $(0, 0)$: $0 < -\frac{1}{2}(0) + 3 \Rightarrow 0 < 3$ TRUE! = $(0, 0)$ satisfies inequality

Shade the solution region: Shade the side containing $(0, 0)$ = Shade BELOW the line

Answer: Dashed line with y-intercept 3 and slope $-\frac{1}{2}$, shaded below.

When the Test Point is on the Line

Graph $y \geq 2x$ on a coordinate plane.

Identify the boundary line: $y = 2x$ passes through the origin with slope 2 = Boundary passes through $(0, 0)$

Determine line style: Symbol is $\geq$, so use a solid line = SOLID line

Choose a different test point: $(0, 0)$ is ON the line, so use $(1, 0)$ instead = Test: $(1, 0)$

Test the point: $0 \geq 2(1) \Rightarrow 0 \geq 2$ FALSE! = $(1, 0)$ is NOT a solution

Shade the opposite side: Since $(1, 0)$ failed, shade the OTHER side (above the line) = Shade ABOVE the line

Answer: Solid line through origin with slope 2, shaded above the line.

Mistake: Using the wrong line style (solid vs. dashed)

Why: Students forget that $<$ and $>$ require a dashed line because points ON the line are not solutions.

Correct: Solid line for $\leq$ and $\geq$ (line included). Dashed line for $<$ and $>$ (line excluded).

Mistake: Shading the wrong region

Why: Students shade above for 'greater than' without testing, but the inequality might be in standard form.

Correct: ALWAYS test a point! Substitute a simple point like $(0, 0)$ to verify which side to shade.

Mistake: Forgetting to flip the inequality when multiplying by negative

Why: When rearranging to slope-intercept form, dividing by a negative number reverses the inequality.

Correct: If you multiply or divide by a negative number, reverse the inequality symbol: $-2y > 6$ becomes $y < -3$.

Mistake: Testing a point that lies on the boundary line

Why: Points on the line don't help determine which side to shade.

Correct: Choose a test point clearly NOT on the line. Origin $(0, 0)$ is ideal unless the line passes through it.

Budget Constraints

When planning expenses, you often have a maximum amount you can spend.

If concert tickets cost 75 dollars and food costs 25 dollars per person, and you have 300 dollars total, then $75t + 25f \leq 300$ where $t$ = tickets, $f$ = food purchases.

Manufacturing Limits

Factories have constraints on production capacity and resources.

A factory can produce at most 500 items per day. If they make chairs and tables, and tables take twice as long, then $c + 2t \leq 500$.

Q: How do I know which side to shade?

A: Pick a test point not on the line ($(0,0)$ is easiest). Substitute into the original inequality. If TRUE, shade that side. If FALSE, shade the opposite side.

Q: Why is the line sometimes dashed?

A: A dashed line means points ON the line are NOT solutions. This happens with strict inequalities ($<$ or $>$). Solid lines ($\leq$ or $\geq$) include points on the line.

Q: What if the origin is on the boundary line?

A: Choose a different test point like $(1, 0)$ or $(0, 1)$. Any point clearly not on the line works.