Graphing Inequalities
An inequality is mathematical statement expressing how two quantities are related. We can represent solutions to both one and two-variable inequalities graphically.
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Graphing One-Variable Inequalities
We can graph inequalities with one variable on a number line. We use a closed dot, \(\bullet,\) to represent \(\leq\) and \(\geq.\) We use an open dot, \(\circ,\) to represent \(<\) and \(>.\)
If \(x \geq -1,\) the we know that \(x\) could be any number that is greater than or equal to -1. We show this solution on a number line by placing a closed dot at -1, to indicate that -1 is a solution, and shade all values on the number line that are greater than -1.
If \(2 \leq y <8,\) then we know that \(y\) can be any number greater than or equal to 2 and less than 8. We show this solution on a number line by placing a closed dot at 2 to indicate that 2 is a solution, placing an open dot at 8 to indicate that 8 is not a solution, and shading all values on the number line between 2 and 8.
Write the inequality that is represented by the number line below.
The graph shows all numbers between -5 and 1, but not including -5 or 1. Therefore, the inequality represented by the graph is \(-5<x<1.\)
Graphing Two-Variable Linear Inequalities
The solution set to a two-variable linear inequality is shown as a shaded graph on the coordinate plane. Shaded regions show the areas that contain points in the solution. If a line is solid, then the points on the line are contained in the solution. If a line is dashed, then the points on the line are not contained in the solution.
To graph a two-variable linear inequality:
Put the inequality into slope-intercept form.
Graph the line that bounds the inequality. Use a solid line for \(\leq\) or \(\geq\) and a dotted line for \(<\) or \(>.\)
Shade above the line for \(>\) or \(\geq.\) Shade below the line for \(<\) or \(\leq.\)
Write an inequality to describe the graph.
The dotted line has a slope of \(-\frac{1}{2}\) and and a \(y\)-intercept of 1, so the equation of the line is \(y=-\frac{1}{2}x+1.\)
The line is dotted, so the solution set does not include the values on on the line.
In addition, the region below the line is shaded, indicating all of the \(y\) values below the line. Therefore, \(y < -\frac{1}{2}x+1.\)
Graphing Systems of Linear Inequalities
When we graph systems of linear inequalities, we graph one inequality at a time. The solution is the shaded region that is the intersection of the inequalities.
\[\begin{cases} y \le 2x+3 \\ y > -\frac{1}{3}x-1 \end{cases}\]
Which shaded region represents the solutions to both of these inequalities?
\[y > x \\ y < - 3x + 4\]
Notice how the two lines split the graph into four regions. The set of solutions that satisfy both inequalities will always be one of these four regions.
For this problem, the solutions to \(y > x\) will be above the line, in regions 1 or 2. The solutions to \(y < -3x + 4\) will be below the line, in regions 2 or 3.
Therefore, the regions that represents the solution set for both inequalities is 2.
For this set of inequalities, where on the coordinate plane do we find points that are solutions to both inequalities?
\[y\leq x + 2 \\ y \geq x + 2\]
It is not possible for a \(y\) value to be both greater than \(x+2\) and less than \(x+2.\) Therefore, the only possible solutions are on the line of \(y=x+2.\)