Inequality

You might find it easier if you swap the z for an x and the swap it back at the end

$(4-z)(z+7) ≥ 2\\ (4-x)(x+7) ≥ 2\\ -x^2 -3x +28 \ge 2\\ -x^2 -3x +26 \ge 0\\ x^2+3x -26 \le 0\\ $

 $y=x^2+3x-26\\ \text{Is a concave up parabola}$

So it will be less than 0 between the roots.  Draw it to see what i mean

Use the quadratic formula to find the roots

etc