Hello, Alejandra9!
Which of the following inequalities has a solution set that, when graphed on the number line,
is a single line segment of finite length?
\(\displaystyle a)\:x^4\,\geq\,1\;\;\;b)\:x^3\,\leq\,27\;\;\;c)\;x^2\,\geq\,16\;\;\;d)\;2\,\leq\,|x|\,\leq\,5\;\;\;e)\;2\,\;\leq\,3x\,+\,4\,\leq\,6\)
. .
\(\displaystyle a)\;x^4\,\geq\,1\;\;\Rightarrow\;\;x^2\,\geq\,1\;\;\Rightarrow\;\;|x|\,\geq\,1\;\;\Rightarrow\;\;x\,\leq -1\,\)
or \(\displaystyle \,x\,\geq\,1\)
. . . = = = = \(\displaystyle \bullet\) - - - - - \(\displaystyle \bullet\) = = = =
. . . . . . . . . -1
. . . . . . 1
\(\displaystyle b)\;x^3\,\leq\,27\;\;\Rightarrow\;\;x\,\leq\,3\)
. . . = = = = = = = = \(\displaystyle \bullet\) - - - - -
. . . . . . . . . . . . . . . . 3
\(\displaystyle c)\;x^2\,\geq\,16\;\;\Rightarrow\;\;|x|\,\geq\,4\;\;\Rightarrow\;\;x\,\leq -4\,\)
or \(\displaystyle \,x\,\geq\,4\)
. . . = = = \(\displaystyle \bullet\) - - - - - \(\displaystyle \bullet\) = = =
. . . . . . . .-4
. . . . . .4
\(\displaystyle d)\;2\,\leq\,|x|\,\leq\,5\)
This has two statements: \(\displaystyle \:x\,\leq\,-2\,\) or \(\displaystyle \,x\,\geq\,2\)
. . . . . . . . . . . . . . . . .and: -\(\displaystyle 5\,\leq\,x\,\leq 5\)
. . . - - \(\displaystyle \bullet\) = = \(\displaystyle \bullet\) - - - \(\displaystyle \bullet\) = = \(\displaystyle \bullet\) - - -
. . . . .-5
. . . -2
. . . 2
. . . .5
\(\displaystyle e)\;2\:\leq\:3x\,+\,4\:\leq\:6\)
Solve for \(\displaystyle x\).
Subtract 4: \(\displaystyle \;-2\:\leq\:3x\:\leq\:2\)
Divivde by 3: \(\displaystyle \;-\frac{2}{3}\:\leq\:x\;\leq\:\frac{2}{3}\)
. . . - - - \(\displaystyle \bullet\) = = = = = \(\displaystyle \bullet\) - - -
. . . . . . -\(\displaystyle \frac{2}{3}\)
. . . . . . . . \(\displaystyle \frac{2}{3}\;\;\;\;\;\)
<----- This one!
