Linear Inequalities

When you replace the equals sign in a linear equation by one of the inequality symbols, you now have a linear inequality. Examples: 1 2 y > x + 1 2x – 3y < 12

Any ordered pair of numbers that satisfies the inequality is a solution of the inequality. Just like an inequality in one variable, linear inequalities can have infinitely many solutions.

y < x + 1 Run 1 y-intercept Rise 1 b = 1 slope m = 1

y < x + 1 y = 2x + 1 Shade TRUE 0 < 0 + 1 0 < 1 Now for the shading . . . Pick a point on either side of the graph. Let’s try (0, 0): Shade Does the point satisfy the inequality? 0 < 0 + 1 TRUE Therefore, shade the half-plane with that point. 0 < 1

y < x + 1 y = 2x + 1 Don't Shade Shade FALSE 2 < - 3 + 1 What if we picked a point on the other side of the line? Now try (-3, 2): Does the point satisfy the inequality? Shade 2 < - 3 + 1 2 < - 2 FALSE Therefore, shade the half-plane opposite that point.

4x - 3y > 12 find the x value when y = 0 find the y value x-intercept 4x - 3(0) = 12 4x - 0 = 12 4x = 12 x = 3 find the y value when x = 0 y-intercept 4(0) - 3y = 12 0 - 3y = 12 -3y = 12 y = -4

4x - 3y > 12 Don't Shade Shade FALSE 4(0) - 3(0) > 12 Now for the shading . . . Pick a point on either side of the graph. Don't Shade Let’s try (0, 0): Does the point satisfy the inequality? Shade FALSE 4(0) - 3(0) > 12 0 - 0 > 12 Therefore, shade the other half-plane opposite the point. 0 > 12