Linear Inequalities is an inequality that represents a half-plane divided by a straight line .
Represents a straight line in a Cartesian coordinate system.
Involves variables raised only to the first power (no exponents higher than 1).
Uses inequality signs such as < , > , <= , or >= to relate the two expressions.
Examples
y / 2 > 8 y/2 > 8 y /2 > 8 2 y < 2 x + 8 2y < 2x + 8 2 y < 2 x + 8 10 x ≥ 50 10x \ge 50 10 x ≥ 50 7 x + 4 ≤ 3 ( 3 x − 8 ) 7x + 4 \le 3(3x - 8) 7 x + 4 ≤ 3 ( 3 x − 8 )
x + 3 < 3 x − 1 x + 3 < 3x - 1 x + 3 < 3 x − 1
x + 3 < 3 x − 1 x + 3 − 3 < 3 x − 1 − 3 x < 3 x − 4 1 x − 3 x < 3 x − 3 x − 4 − 2 x < − 4 − 2 x × − 1 < − 4 × − 1 2 x > 4 2 x 2 > 4 2 2 x 2 > 2 x > 2 x + 3 < 3x - 1\\
x + 3 - 3 < 3x - 1 - 3\\
x < 3x - 4\\
1x - 3x < 3x - 3x - 4\\
-2x < -4\\
-2x \times -1 < -4 \times -1\\
2x > 4\\
\frac{2x}{2} > \frac{4}{2}\\
\frac{\cancel{2}x}{\cancel{2}} > 2\\
x > 2 x + 3 < 3 x − 1 x + 3 − 3 < 3 x − 1 − 3 x < 3 x − 4 1 x − 3 x < 3 x − 3 x − 4 − 2 x < − 4 − 2 x × − 1 < − 4 × − 1 2 x > 4 2 2 x > 2 4 2 2 x > 2 x > 2
2 ( x + 2 ) − 1 ≤ 5 + 2 ( 4 − x ) 2(x + 2) - 1 \le 5 + 2(4 - x) 2 ( x + 2 ) − 1 ≤ 5 + 2 ( 4 − x )
2 ( x + 2 ) − 1 ≤ 5 + 2 ( 4 − x ) 2 x + 4 − 1 ≤ 5 + 8 − 2 x 2 x + 3 ≤ 13 − 2 x 2 x + 3 − 3 ≤ 13 − 3 − 2 x 2 x ≤ 10 − 2 x 2 x + 2 x ≤ 10 − 2 x + 2 x 4 x ≤ 10 4 x 4 ≤ 10 4 4 x 4 ≤ 2.5 x ≤ 2.5 2(x + 2) - 1 \le 5 + 2(4 - x)\\
2x + 4 - 1 \le 5 + 8 - 2x\\
2x + 3 \le 13 - 2x\\
2x + 3 - 3 \le 13 - 3 - 2x\\
2x \le 10 - 2x\\
2x + 2x \le 10 - 2x + 2x\\
4x \le 10\\
\frac{4x}{4} \le \frac{10}{4}\\
\frac{\cancel{4}x}{\cancel{4}} \le 2.5\\
x \le 2.5 2 ( x + 2 ) − 1 ≤ 5 + 2 ( 4 − x ) 2 x + 4 − 1 ≤ 5 + 8 − 2 x 2 x + 3 ≤ 13 − 2 x 2 x + 3 − 3 ≤ 13 − 3 − 2 x 2 x ≤ 10 − 2 x 2 x + 2 x ≤ 10 − 2 x + 2 x 4 x ≤ 10 4 4 x ≤ 4 10 4 4 x ≤ 2.5 x ≤ 2.5
The best way to graph a linear inequality is to write the inequality in slope-intercept form , then test the inequality to determine which side of the line to shade.
The slope-intercept formula
Depending on the inequality operator used, the line in the graph will be solid or dashed.
> or < - dashed.≥ \ge ≥ ≤ \le ≤ Graphing Inequalities (youtube) Graphing two variable inequality (youtube)
2(x + 10) ≤ \le ≤ 2 ( x + 10 ) ≤ 30 2 x + 20 ≤ 30 y ≥ 2 x + 20 2(x + 10) \le 30\\
2x + 20 \le 30\\
y \ge 2x + 20 2 ( x + 10 ) ≤ 30 2 x + 20 ≤ 30 y ≥ 2 x + 20
6x + 3y > 9 6 x + 3 y > 9 6 x − 6 x + 3 y > − 6 x + 9 3 y > − 6 x + 9 3 y 3 > − 6 x + 9 3 y > − 2 x + 3 6x + 3y > 9\\
6x - 6x + 3y > -6x + 9\\
3y > -6x + 9\\
\frac{\cancel{3}y}{\cancel{3}} > \frac{-6x+9}{3}\\
y > -2x + 3 6 x + 3 y > 9 6 x − 6 x + 3 y > − 6 x + 9 3 y > − 6 x + 9 3 3 y > 3 − 6 x + 9 y > − 2 x + 3
3(y + 3) < 6(x - 2) + 12 3 ( y + 3 ) < 6 ( x − 2 ) + 12 3 y + 9 < 6 x − 12 + 12 3 y + 9 < 6 x 3 y + 9 − 9 < 6 x − 9 3 y < 6 x − 9 3 y 3 < 6 x − 9 9 y < 2 x − 3 3(y + 3) < 6(x - 2) + 12\\
3y + 9 < 6x - 12 + 12\\
3y + 9 < 6x\\
3y + 9 - 9 < 6x - 9\\
3y < 6x - 9\\
\frac{\cancel{3}y}{\cancel{3}} < \frac{6x-9}{9}\\
y< 2x - 3 3 ( y + 3 ) < 6 ( x − 2 ) + 12 3 y + 9 < 6 x − 12 + 12 3 y + 9 < 6 x 3 y + 9 − 9 < 6 x − 9 3 y < 6 x − 9 3 3 y < 9 6 x − 9 y < 2 x − 3
-2x - 2y ≥ \ge ≥ − 2 x − 2 y ≥ 2 x + 4 − 2 x + 2 x − 2 y ≥ 2 x + 2 x + 4 − 2 y ≥ 4 x + 4 − 1 × − 2 y ≥ − 1 ( 4 x + 4 ) 2 y ≤ − 4 x − 4 2 y 2 ≤ − 4 x − 4 2 y ≤ − 2 x − 2 -2x - 2y \ge 2x + 4\\
-2x + 2x - 2y \ge 2x + 2x + 4\\
-2y \ge 4x + 4\\
-1 \times -2y \ge -1(4x+4)\\
2y \le -4x - 4\\
\frac{\cancel{2}y}{\cancel{2}} \le \frac{-4x-4}{2}\\
y \le -2x - 2 − 2 x − 2 y ≥ 2 x + 4 − 2 x + 2 x − 2 y ≥ 2 x + 2 x + 4 − 2 y ≥ 4 x + 4 − 1 × − 2 y ≥ − 1 ( 4 x + 4 ) 2 y ≤ − 4 x − 4 2 2 y ≤ 2 − 4 x − 4 y ≤ − 2 x − 2
