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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)

Example 1

Write the inequality for the given graph.

$$\begin{gathered} a=1 \\ h=5 \\ k=0 \\ \mathrm{symbol}: > \\ y=a\sqrt{x–h}+k \\ y>\sqrt{–\left(x–5\right)} \\ y>\sqrt{–x+5} \end{gathered}$$

Determine if $\left\{\mathit{A}\mathbf{(}\mathbf{–}\mathbf{2}\mathbf{,}\mathbf{ }\mathbf{–}\mathbf{3}\mathbf{)}\mathbf{,}\mathbf{ }\mathit{B}\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{ }\mathbf{3}\mathbf{)}\mathbf{,}\mathbf{ }\mathit{C}\mathbf{(}\mathbf{5}\mathbf{,}\mathbf{ }\mathbf{–}\mathbf{1}\mathbf{)}\mathbf{,}\mathbf{ }\mathit{D}\mathbf{(}\mathbf{5}\mathbf{,}\mathbf{ }\mathbf{0}\mathbf{)}\right\}$ 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   .

Example 2

Write the inequality for the given graph.

$$\begin{gathered} a=–\frac{1}{2} \\ h=4, k=1 \\ \mathrm{symbol}: < \\ y=a|x–h|+k \\ y<–\frac{1}{2}|x–4|+1 \end{gathered}$$

Example 3

Write the system of inequalities for the given graph.