Linear Inequalities
A linear inequality is an inequality of a line. An example would be \(y>2x+3\). In this graph, all of the coordinates that would satisfy the inequality are shaded.
If we were asked to solve and graph an inequality, the \(y\) might be replaced with \(0\), or any other value or expression. When we only have one variable in the inequality, we can solve for that value by isolating the variable.
\(0>2x + 3\)
\(-3>2x\)
\(-\frac{3}{2}<x\)
To graph this on a number line, we'd graph \(-\large\frac{3}{2}\), decide whether to plot an open (exclusive of the value) or closed (inclusive of the value) dot, and decide which way to shade.
A linear inequality is an inequality of a line. An example would be \(y>2x+3\). In this graph, all of the coordinates that would satisfy the inequality are shaded.
If we were asked to solve and graph an inequality, the \(y\) might be replaced with \(0\), or any other value or expression. When we only have one variable in the inequality, we can solve for that value by isolating the variable.
\(0>2x + 3\)
\(-3>2x\)
\(-\frac{3}{2}<x\)
To graph this on a number line, we'd graph \(-\large\frac{3}{2}\), decide whether to plot an open (exclusive of the value) or closed (inclusive of the value) dot, and decide which way to shade.
When solving inequalities, two things to look out for are inequalities that have no solution or infinitely many solutions.
Example 1: Solve the inequality: \(2x+6>2x-8\).
By subtracting \(2x\) from both sides, we get \(6>-8\). This is a true statement, therefore this problem has infinitely many solutions. If we were to graph the solution, we'd shade the whole number line.
Example 1: Solve the inequality: \(2x+6>2x-8\).
By subtracting \(2x\) from both sides, we get \(6>-8\). This is a true statement, therefore this problem has infinitely many solutions. If we were to graph the solution, we'd shade the whole number line.
Example 2: Solve the inequality: \(5\left(2x-7\right)>7x+3\left(x-11\right)\).
Upon distributing and simplifying each side of the inequality, we get \(10x-35>10x-33\). Since \(-35\) is not greater than \(-33\), this is a false statement, therefore, there is no solution. In order to graph this solution, you'd draw a number line and label some values, but you wouldn't shade the number line because no values make this a true inequality.