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Graphing inequalities is an essential part of understanding algebraic concepts. It helps in visualizing the solutions to inequality equations on a number line or coordinate plane. Unlike equations, inequalities do not have just one solution, but rather a range of solutions.
What are Inequalities?
Inequalities are mathematical expressions involving the symbols < (less than), <= (less than or equal to), > (greater than), or >= (greater than or equal to). These symbols are used to compare two values or expressions. For example, \( x > 5 $\)means that x is greater than 5.
Graphing on a Number Line
Graphing an inequality on a number line gives a visual representation of all its possible solutions.
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To graph \( x > 5 \), we would draw an open circle at 5 and shade the line to the right of 5, indicating all the numbers greater than 5.
library(ggplot2) ggplot(data.frame(x = c(0, 10)), aes(x)) + geom_segment(aes(xend = 10, y = 0, yend = 0)) + geom_point(aes(x = 5, y = 0), shape = 1) + geom_segment(aes(x = 5.1, xend = 10, y = 0, yend = 0), arrow = arrow(type = "open"))
Graphing on a Coordinate Plane
Inequalities with two variables, like \( y > 2x + 3 \), are graphed on a coordinate plane.
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First, graph the equation \( y = 2x + 3 $\)as if it were an equality.
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Since the inequality is \( y > 2x + 3 \), shade above the line to represent all the points where y is greater than \( 2x + 3 \).
library(ggplot2) ggplot(data.frame(x = c(-10, 10)), aes(x)) + stat_function(fun = function(x) 2x + 3) + geom_ribbon(aes(ymin = 2x + 3, ymax = 10), fill = "blue", alpha = 0.2)
Practice Problems
Try graphing the following inequalities:
- \( x <= 4 $\)on a number line.
- \( y < -x + 2 $\)on a coordinate plane.
Graphing inequalities is not just about drawing lines and shading areas; it represents a fundamental understanding of algebra and the solutions to inequalities. Practice is key to mastering this concept.