Graphing Inequalities: Practice

Substitute $x = 0$ and $y = 0$ into the inequality and check the result.

$0 < 0 + 4 \;\Rightarrow\; 0 < 4$

That statement is true, so the origin is a solution. It is not on the boundary $y = x + 4$, which passes through $(0, 4)$.

A true result at a test point means that point is a solution, so its whole side is shaded.

$0 > 0 - 2 \;\Rightarrow\; 0 > -2 \ \text{(true)}$

Since the origin satisfies the inequality, shade the half-plane that contains it.

A false result at the test point means the origin is not a solution, so the solutions are on the opposite side.

$4(0) + 0 < -1 \;\Rightarrow\; 0 < -1 \ \text{(false)}$

Shade the half-plane that does not contain the origin.

An inequality in $x$ alone has a vertical boundary $x = 3$, and $x > 3$ means the $x$-coordinate is larger than $3$.

$x > 3 \;\Rightarrow\; \text{shade where } x \text{ is bigger, to the right}$

Larger $x$ lies to the right, so shade right of the line.

An inequality in $y$ alone has a horizontal boundary $y = -2$, and $y \le -2$ means the height is at most $-2$.

$y \le -2 \;\Rightarrow\; \text{shade where } y \text{ is smaller, below}$

The symbol is inclusive, so the line is included and you shade on or below it.

The inequality involves $x$ alone, so the boundary $x = 1$ is vertical, and $\ge$ is inclusive, so the line is solid.

$x = 1 \ \text{is vertical}, \quad \ge \;\Rightarrow\; \text{solid}$

A vertical line drawn solid matches both facts.

The boundary is the vertical line $x = 2$, and it is solid, so the symbol is inclusive. The shaded region is to the left, where $x$ is smaller.

$\text{solid vertical line at } x = 2,\ \text{shaded left} \;\Rightarrow\; x \le 2$

The origin is shaded, and $0 \le 2$ is true, confirming the choice.

Every pair $(x, y)$ that makes the inequality true is a solution, and those pairs fill one whole side of the boundary line.

$y > mx + b \ \text{holds on one entire side of}\ y = mx + b$

That region is called a half-plane.

Substitute the point into $y \ge x$.

$1 \ge 1$

That is true, so $(1, 1)$ is a solution. It lies exactly on the boundary $y = x$, and since $\ge$ is inclusive, that boundary is solid and its points count.

The boundary comes from replacing the inequality symbol with an equals sign, changing nothing else.

$5x - 2y \ge 10 \;\Rightarrow\; 5x - 2y = 10$

That line is the border of the solution region.

A dashed line marks a strict inequality ($<$ or $>$), for which equality is not allowed.

$\text{dashed} \;\Leftrightarrow\; \text{strict } (<,\,>) \;\Rightarrow\; \text{line excluded}$

So the points exactly on a dashed line are not part of the solution.

The origin makes the arithmetic trivial, and it works whenever the boundary does not pass through it. Here $0 = 3(0) - 4$ is $0 = -4$, false, so the line misses the origin.

$\text{test } (0,0):\ 0 < 3(0) - 4 \;\Rightarrow\; 0 < -4$

So $(0, 0)$ is a valid and easy test point.