Draw the graphs of y= 13(4x+3) and y= 23x+2. When does the first graph lie above the second graph?
{x|x ≥ 32 }
Practice makes perfect
To solve the given inequality using a graphing calculator, we need to treat each side of the inequality symbol as its own individual equation.
1/3(4x+3) ≥2/3x+2
⇓ ly=1/3(4x+3) y=2/3x+2
⇓
ly=4/3x+1 y=2/3x+2
Then we can look for the values of x where the value of 43x+1 is greater than or equal to the value of 23x+2. In other words, we want to find the values of x where the graph of y= 43x+1 lies above the graph of y= 23x+2.
Graphing the Functions
Press Y= on the calculator to access the screen to input our functions.
To draw the graph, press the GRAPH button.
Finding the Solution Set
We see that the graphs intersect at a point. The x-coordinate of this point can be found by pressing 2ND and then CALC.
After selecting the intersect option, we choose the first and second curves. Then, the calculator asks for a guess where the intersection point might be. After that, it will calculate the exact point for us.
The lines intersect at x=1.5, or x= 32. Tracing the lines on the calculator screen using the TRACE button, we can see that y= 43x+1 is the upper line after the intersection. Since the inequality is non-strict, it is also true at the point of intersection. This means that the given inequality holds true when x≥ 32.
4/3x+1≥2/3x+2
⇔
x≥1.5
We can now write the solution set for our inequality.
{x|x ≥ 32 }
