Sketch, on a pair of axes, the curve with equation y = 6 - |3x+4| , indicating the coordinates where the curve crosses the axes, then solve the equation x = 6 - |3x+4|
One helpful method for solving questions like these is to sketch the curve in stages. First begin with the straight line y=3x+4 (taking 3x+4 from within the modulus lines) crossing the x axis at -4/3 and the y axis at 4. Next, move on to y=|3x+4|. This should look the same at all points to the right of the x intercept/above the x axis - however, all points to the left of x=-4/3 should now have been reflected in the x axis, creating a V shaped curve. Now, move on to y=-|3x+4|. This should be the previous curve, reflected in the x axis - upside down, with the point of the (now upside down) V still at x=-4/3. Finally, we consider y=6-|3x+4|. The entirety of the upside down V from the previous curve should now be 6 higher on the y axis. We can find the new x-intercepts by solving 0=6 +/- (3x+4), which gives us the upside down V shape with x-intercepts x=-10/3 and x=2/3, y-intercept at y=2, and the point of the V at (-4/3,6).
To solve x=6-|3x+4|, solve the equations x=6-(3x+4) and x=6--(3x+4). The first gives us x=6-3x-4=2-3x, which rearranges to 4x=2, and so x=1/2. The second gives us x=6+3x+4=10+3x, which rearranges to -2x=10, and so x=-10/2=-5.
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