Ulikheter
Oppgave 1
-
\begin{align*}
x^2+x&<2\\
x^2+x-2&<0\\
(x+2)(x-1)&<0\\
x&\in\big< -2,1\big>
\end{align*} -
\begin{align*}
1-x^2&<-8\\
-x^2+1+8&<0\\
x^2-9&>0\\
(x+3)(x-3) &>0\\
x&\in \big< \leftarrow,-3 \big>\cup \big<3, \rightarrow \big>
\end{align*} -
\begin{align}
x^2 &\geq 4x\\
x^2-4x &\geq 0\\
x(x-4) &\geq 0\\
x&\in \big< \leftarrow, 0\big] \cup \big[4,\rightarrow\big>
\end{align}
Oppgave 2
-
\begin{align*}
x^2-4x-12 &< 0\\
(x-6)(x+2) &< 0\\x&\in \big< -2,6\big>
\end{align*} -
\begin{align*}
-x^2-x+6 &\geq 0\\
x^2+x-6 &\leq 0\\
(x+3)(x-2) &\leq 0\\
x&\in \big[ -3,2 \big]
\end{align*} -
\begin{align*}
x^2-8x &\leq-15\\
x^2-8x+15 &\leq 0\\
(x-5)(x-3) &\leq 0\\
x&\in \big[ 3, 5\big]
\end{align*}
Oppgave 3
-
\begin{align*}
1 &>x^2\\
x^2-1 &< 0\\
(x+1)(x-1) &< 0\\
x&\in \big< -1, 1\big>
\end{align*} -
\begin{align*}
-x &\leq -x^2+6\\
x^2-x-6 &\leq 0\\
(x-3)(x+2) &\leq 0\\
x&\in \big[ -2, 3\big]
\end{align*} -
\begin{align*}
\frac{1}{2}\Big(\frac{1}{x}+3\Big) &<1-\frac{1}{x}\\
\frac{1}{2x}+\frac{3}{2} &< 1-\frac{1}{x}\\
\frac{1}{2x}+\frac{3x}{2x}&< \frac{2x}{2x}-\frac{2}{2x}\\
\frac{1+3x-2x+2}{2x} &<0\\
\frac{x+3}{2x} &< 0\\
x&\in \big< -3, 0\big>
\end{align*}
Oppgave 4
-
\begin{align*}
x-4x^2 &\leq 0\\
4x^2-x &\geq 0\\
4x(x-\frac{1}{4}) &\geq 0\\
x &\in \big< \leftarrow,0\big>\cup\bigg<\frac{1}{4},\rightarrow\bigg>
\end{align*} -
\begin{align*}
2x^2+5x+3 &>0\\
2(x+3)\Big(x-\frac{1}{2}\Big)&>0\\
x &\in \big<\leftarrow,-3\big>\cup \big<-\frac{1}{2},\rightarrow\big>
\end{align*} -
\begin{align*}
-3x^2+27 &>0\\
-3(x^2-9) &> 0\\
(x+3)(x-3) &< 0\\
x&\in \big<-3,3\big>
\end{align*}
Oppgave 5
-
\begin{align*}
1-2x &\geq -x^2\\
x^2-2x+1 &\geq 0\\
(x-1)^2 &\geq 0\\
x&\in \mathbb{R}
\end{align*}