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Solutions to Inequalities Solutions

When given an inequality or system of inequalities, use substitution to check algebraically if an ordered pair is or is not a solution.
If you have a graph of an inequality, estimate the location of the ordered pair on the graph.
Remember, these items are excluded from inequality solutions:
- dashed curves
- open points
- asymptotes
To write an inequality from a graph consider these items:
- the family to which it belongs (use your formula sheet)
- how it is transformed , or the values a, h, k
- the inequality symbol (solid or dashed curve and shading)

Write the inequality for the given graph.

Plan

Square root
Reflected over the y-axis: f(–x)
Translated 5 to the right
Shading above, dashed

$a = 1 h = 5 k = 0 symbol : > y = a \sqrt{x - h} + k y > \sqrt{- (x - 5)} y > \sqrt{- x + 5}$

Determine if $\{A (- 2, - 3), B (1, 3), C (5, - 1), D (5, 0)\}$ are solutions to the inequality.

Point A is not a solution because it is not found within the shaded region .

Point B is a solution because it is found in the shaded region .

Point C is not a solution because it is not found within the shaded region .

Point D is not a solution because it is on the dashed line .

Write the inequality for the given graph.

Plan

Absolute value
Reflected over the x-axis
Translated right 4 and up 1
Shading below, dashed

$a = - \frac{1}{2} h = 4, k = 1 symbol : < y = a | x - h | + k y < - \frac{1}{2} | x - 4 | + 1$

Write the system of inequalities for the given graph.

$Cubic : \geq a = 1, h = 0, k = 0 y = a {(x - h)}^{3} + k Quadratic : < a = - 1, h = 0, k = 3 y = a {(x - h)}^{2} + k y \geq x^{3} y < - x^{2} + 3$