Compound Inequalities Worksheets

When two inequalities are considered, together they form compound inequalities. Compound inequalities may come in two forms: “AND compound inequalities” and “OR compound inequalities”. The inequalities, “\(2x < 3\) and \(x \geq 3 \)” and “\(2x < 3\) or \(x \geq 3\)” are examples of compound inequalities. The solutions to compound inequalities that hold “AND” refer to those values of the input variable, x that make both inequalities true. For instance, the solution to “\(2x < 3\) and \(x \geq 3 \)” are those numbers that can satisfy both inequalities: “\(2x < 3, x \geq 3 \)”. On the other hand, the solutions to compound inequalities joined by the word “OR” refer to those values of its input variable that can satisfy either the first inequality or the second one.

The graph of the solution set of a compound inequality with the word “AND” is the intersection of the individual solution set of both inequalities while the graph of the solution set of a compound inequality joined by the word “OR” is the union of the individual solution set of the two inequalities it comprised. To look for the solution set of any compound inequality, we solve individually each inequality involved then get the intersection of their solution set if it contains “AND”. Otherwise, get the union if it contains the word “OR”.

EXAMPLES:

Solve each compound inequality. Sketch the graph and write the solution set.

\( \begin{align*} &1.\;\;\; 2x < 8\; \text{and}\;x\geq 2\\&2.\;\;\;3k-3>3\; \text{or}\; 2k+1\geq -3\\&3.\;\;\;3k-3>3\;\text{and}\;2k+1\geq -3\\&4.\;\;\;3x-4>5\;\text{or}\; 2\left ( x-1 \right )>x-2 \end{align*}\)

SOLUTIONS:

\(\begin{align*} 1.\;\;\; \displaystyle &2x < 8\; \text{and}\;x\geq 2\\&\frac{2x}{2}<\frac{8}{2}\;\text{and}\;x\geq 2\\&x<4\;\text{and}\;x\geq 2\\ \\ &\textbf{Solution set:}\;\left ( -\infty,4 \right )\cap \left [2,+\infty \right )=\left [ 2,4 \right ) \end{align*}\)

\(\begin{align*}2.\;\;\; &3k-3>3\;\text{or}\;2k+1\geq -3\\&3k > 6\;\text{or}\;2k\geq -4\\&k>2\;\text{or}\;k\geq -2\\\\&\textbf{Solution set:}\;\left ( 2,+\infty \right )\cup\left ( -\infty,-2 \right ] \end{align*}\)

\(\begin{align*}3.\;\;\; &3k-3>3\;\text{and}\;2k+1\geq -3\\&3k > 6\;\text{and}\;2k\geq -4\\&k>2\;\text{and}\;k\geq -2\\\\&\textbf{Solution set:}\;\left ( 2,+\infty \right )\cup\left ( -\infty,-2 \right ] =\emptyset \end{align*}\)
Graph: NO graph

\(\begin{align*}4.\;\;\; &3x-4>5\quad \text{or}\quad \;2\left ( x-1 \right )>x-2\\&3x > 9\quad \text{or}\quad 2x-2>x-2 \\&x>3\quad\text{or}\quad x>0\\\\&\textbf{Solution set:}\;\left ( 3,+\infty \right )\cup\left ( 0,+\infty \right ) =\left ( 3,+\infty \right ) \end{align*}\)