Find the solution set of the linear inequalities shown below.
An inequality is like an equation, except that it doesn’t use an equal sign. It uses a different kind of symbol; it could be a less than, greater than, less than or equal to, greater than or equal to — one of those four. So before we decide which symbol we will replace the equal sign, let’s first find the equations of both of those lines and then decide what symbol to use.
The equation of a line is 𝑦 equals 𝑚𝑥 plus 𝑏. 𝑚 is the slope of the graph, which is like the steepness of the graph, and 𝑏 is the 𝑦-intercept, where it crosses the 𝑦-axis. Let’s begin with this line. Where does it cross the 𝑦-axis? It crosses at zero. So we can take the equation number line and replace 𝑏 with zero.
Next is the slope. There is an equation for slope, but since we actually have a graph, we can count. Slope is the vertical change divided by the horizontal change. In verticals, up and down. And if we go up, that’s positive; if we go down, that’s negative. Horizontals, left and right. If we go right, that’s positive; if we go left, that’s negative. So from our original point at zero, zero, if we go to the right, we’re going down.
So first is the vertical change. We would have to go down one, which is negative for the vertical change. And then we would have to go right two, which is positive two for the horizontal change. And this will stay constant. So we can replace 𝑚 with negative one-half. So that line is 𝑦 equals negative one-half 𝑥 because we don’t have to add the plus zero.
Let’s look at our next line; it’s 𝑦-intercept is at negative five. And then for its slope, the next point that it goes through would have to go down one and right two, so negative one over two. So negative one-half is 𝑚. So if we would simplify that, we would get negative one-half 𝑥 minus five cause plus negative five is just minus five.
So these are inequalities. So they’re really not equal signs; there’s something else. If we would use a less than or a greater than symbol, the lines will be dashed; if we would use a less than or equal to or a greater than or equal to, the lines will be solid. And here, they’re both solid.
So the next step is the shading. Less than or less than or equal to, you would shade below the graph and then greater than or greater than or equal to, you would shade above the graph. So for this red shading, it’s above that line and it’s a solid line. So it would be 𝑦 it’s greater than or equal to negative one-half 𝑥. The blue is below with a solid line, so 𝑦 it’s then less than or equal to negative one-half 𝑥 minus five.
Now, we went through all of this to find the solution, meaning where they overlap. So could we have solve this right from the beginning? Probably, but it’s always a good idea to break down every single piece of our questions to make sure we’re doing everything correctly. So where do these shadings overlap? Nowhere. And since they don’t overlap anywhere, there are no solutions. So our answer is no solution.