Compound inequalities
Two inequalities joined by the word 'or' or the word 'and' form a compound inequality.
Examples of compound inequalities:
$x\le 5 \quad or \quad 5\gt 2x +1$
$8\gt x-2 \quad and \quad x\le 7$
A compound inequality of the form $a\lt x\lt b$ is the same as $a\lt x$ and $x\lt b$. This means $x$ is between $a$ and $b$.
In interval notation, $a\lt x\lt b$ is $(a,b)$ and
$a\le x\le b$ is $[a,b]$.
The word 'or' corresponds to the union operation and the word 'and' corresponds to the intersection operation.
Union
If $A$ and $B$ are two sets, then the union of $A$ and $B$ is the set of elements that are in $A$, in $B$, or in both.
The union of $A$ and $B$, read as "$A$ union $B$", is denoted $A\cup B$.
The union operation combines the two sets.
Example: If $A=\{1,2,3,4,5\}$, and $B=\{3,4,7,8\}$, then
$A\cup B=\{1,2,3,4,5,7,8\}\:$ (combine $A$ and $B$).
Example: Find the union of the sets A and B, where $A=\{x|x\lt -2\}$ and $B=\{x|x\ge 3\}$.
Solution: Draw the graph of $A$ and $B$ and combine them:
Intersection
If $A$ and $B$ are two sets, then the intersection of $A$ and $B$ is the set of elements that are common to both $A$ and $B$.
The intersection of $A$ and $B$, read as "$A$ intersection $B$", is denoted $A\cap B$.
The intersection operation extracts the elements common to both sets.
Example 1: If $A=\{1,2,3,4,5\}$, and $B=\{3,4,7,8\}$, then
$A\cap B=\{3,4\}$, since only $3$ and $4$ are in both $A$ and $B$.
Example 2: If $P=\{a, b, c\}$ and $Q=\{e, f, g\}$, then
$P\cap Q=\{\quad \}$, the empty set. The symbol for the empty set is $\emptyset$.
Example: Find the intersection of the sets A and B, where $A=\{x|x\ge -3\}$ and $B=\{x|x\lt 2\}$.
Solution: Draw the graph of $A$ and $B$ and take the part of the solution common to both:
Solving compound inequalities
To solve a compound inequality, solve each inequality separately. Then take the union of the solutions if they are joined by the word or, or the intersection of the solutions if they are joined by the word and.
Example 1: Solve, $4x\lt x-6$ or $x\ge 3$
Solution:
$\begin{align*} 4x\lt x-6 \, &or\ x\ge 3\\ -x:\quad 4x-x\lt -6 \, &or \, x\ge 3\\ \div 3: \qquad x\lt -2 \, &or\, x\ge 3 \end{align*}$
Solution in graphical form:
Solution in interval notation:
$(-\infty,-2)\cup[3,\infty)$
Example 2: Solve $4x-2\lt 14$ and $2x\ge 2$
Solution:
$\begin{align*} 4x-2\lt 14\quad &and \quad2x\ge 2\\ 4x\lt 16\quad &and \quad x\ge 1\\ x\lt 4\quad &and \quad x\ge 1 \end{align*}$
The solution in the graphical form:
The solution in interval notation:
$[1,4)$