Linear Inequalities
A linear inequality in one variable is a statement that involves one of several inequality symbols as indicated in the diagrams below. A solution to an inequality is every value of the variable that will result in a true statement. We solve these algebraically and also represent the solutions graphically.
Before we proceed with examples, we need to look at the different symbols we use for inequalities, what they mean and what a graph of the solution would look like.
A linear inequality in one variable is a statement that involves one of several inequality symbols as indicated in the diagrams below. A solution to an inequality is every value of the variable that will result in a true statement. We solve these algebraically and also represent the solutions graphically.
Before we proceed with examples, we need to look at the different symbols we use for inequalities, what they mean and what a graph of the solution would look like.
\(x\ne5\) means the solution for \(x\) is all real numbers except \(5\).
\(x<5\) means the solution for \(x\) is every real number less than \(5\) but not including \(5\).
\(x\le5\) means the solution for \(x\) is every real number less than \(5\) and equal to \(5\).
\(x>5\) means the solution for \(x\) is every real number greater than \(5\) but not including \(5\).
\(x\ge5\) means the solution for \(x\) is every real number greater than \(5\) and equal to \(5\).
Notice, when you are graphing \(<\) or \(>\), an open circle is used at the number to indicate that the number is not part of the solution set (we say the solution set is not inclusive of these values). When you are graphing \(\le\) or \(\ge\), a closed circle is used to indicate that the number is part of the solution set (we say the solution set is inclusive of these values).
Let’s look at some basic examples before we move into compound inequalities.
Example 1: Solve and graph the inequality \(3x-7\ge-3+5x\).
\(\begin{align}3x-7&\ge-3+5x \ & \ &\text{1) Rewrite the inequality.}\\-2x-7&\ge-3 \ & \ &\text{2) Subtract 5x from both sides.}\\-2x&\ge4 \ & \ &\text{3) Add 7 to both sides.}\\x&\le-2 \ & \ &\text{4) Divide both sides by -2.}^*\end{align}\)
*Note: When solving an inequality, if you multiply or divide by a negative number, the direction of the inequality sign will change. This does not occur when you add or subtract a value from both sides.
Example 2: Solve and graph the inequality \(\frac{2}{3}\left(x-4\right)<-8\).
Compound Inequalities
A compound inequality consists of two simple inequalities joined by “and” or “or".
Example 3: Graph the solution of \(-1\le x<5\).
\(-1\le x<5\) means the solution for \(x\) is all real numbers that are greater than or equal to \(-1\) AND less than \(5\).
The graph is shown below:
Example 4: Graph the solution of \(x<-4\ \text{or}\ x\ge5\).
\(x<-4\ \text{or}\ x\ge5\) means the solution for \(x\) is all real numbers that are less than \(-4\) OR greater than or equal to \(5\).
The graph is shown below:
To solve compound inequalities, you are really just solving two simple inequalities and writing your answers with the proper notation.A solution to the “or” inequalities will always contain the word “or” between the solutions. A solution to the “and” inequalities will always have the variable between the two numbers, with the smallest number on the left and the largest number on the right--the inequality sign will always point to the left.
Example 5: Solve the inequality \(4<\frac{1}{2}\left(x+3\right)<7\).
This is a compound “and” inequality. You are really solving \(-5\le7-2x\) AND \(7-2x<8\). Usually, we don’t split it into two separate inequalities, we just perform the same operations on the left side and the right side of the statement. However, in compound "and" inequalities, if there are variables on both sides, it is necessary to split the compound inequality into two separate inequalities before solving.
\(\begin{align}4<&\frac{1}{2}\left(x+3\right)<7 \ & \ &\text{1) Rewrite the inequality.}\\8<&x+3<14 \ & \ &\text{2) Multiply both sides by 2.}\\5<&x<11 \ & \ &\text{3) Subtract 3 from both sides.}\end{align}\)
Example 6: Solve the inequality \(-5\le7-2x<8\).
Example 7: Solve the inequality \(-\frac{2}{3}x+1>8\) or \(\frac{1}{5}x-7\ge-5\).