Use the given graphs to solve the following inequalities.
-2x+3≥-1
The solution of an inequality f\left(x\right)\geqslant g\left(x\right) is the interval on the x-axis for which the graph of f is above the graph of g.
In our problem:
- f\left(x\right)=-2x+3
- g\left(x\right)=-1
The interval on the x-axis for which the graph of f is above the graph of g is \left(-\infty,2\left].
The solution set is \left(-\infty,2\left].
3x-6 \lt -x+2
The solution of an inequality f\left(x\right) \lt g\left(x\right) is the interval on the x-axis for which the graph of f is below the graph of g.
In our problem:
- f\left(x\right)=3x-6
- g\left(x\right)=-x+2
The interval on the x-axis for which the graph of f is below the graph of g is \left(-\infty,2\right).
The solution set is \left(-\infty,2\right).
-5x+8 \gt -3x+2
The solution of an inequality f\left(x\right) \gt g\left(x\right) is the interval on the x-axis for which the graph of f is above the graph of g.
In our problem:
- f\left(x\right)=-5x+8
- g\left(x\right)=-3x+2
The interval on the x-axis for which the graph of f is above the graph of g is \left(-\infty,3\right).
The solution set is \left(-\infty,3\right).
2x-4 \leqslant 2x+6
The solution of an inequality f\left(x\right) \leqslant g\left(x\right) is the interval on the x-axis for which the graph of f is below the graph of g.
In our problem:
- f\left(x\right)=2x-4
- g\left(x\right)=2x+6
The interval on the x-axis for which the graph of f is below the graph of g is \left(-\infty,\infty\right), in other words, all real numbers.
The solution set is \mathbb{R}.
4 \geqslant 3x-5
The solution of an inequality f\left(x\right) \geqslant g\left(x\right) is the interval on the x-axis for which the graph of f is above the graph of g.
In our problem:
- f\left(x\right)=4
- g\left(x\right)=3x-5
The interval on the x-axis for which the graph of f is above the graph of g is \left(-\infty,3\left].
The solution set is \left(-\infty,3\left].
-x+7 \lt 2x+1
The solution of an inequality f\left(x\right) \lt g\left(x\right) is the interval on the x-axis for which the graph of f is below the graph of g.
In our problem:
- f\left(x\right)=-x+7
- g\left(x\right)=2x+1
The interval on the x-axis for which the graph of f is below the graph of g is \left(2,\infty\right).
The solution set is \left(2,\infty\right).
\dfrac{1}{3}x+2 \gt -\dfrac{2}{5}x-\dfrac{1}{5}
The solution of an inequality f\left(x\right) \gt g\left(x\right) is the interval on the x-axis for which the graph of f is above the graph of g.
In our problem:
- f\left(x\right)=\dfrac{1}{3}x+2
- g\left(x\right)=-\dfrac{2}{5}x-\dfrac{1}{5}
The interval on the x-axis for which the graph of f is above the graph of g is \left(-3,\infty\right).
The solution set is \left(-3,\infty\right).