1. **Problem:** Given the equation $$24^x - 2 \tan(3x) - 10^x = 14^x$$ with $$x \leq 5 \times 90^\circ$$, find the value of $$x$$ correct to the nearest whole number. 2. **Formula and rules:** This is a transcendental equation involving exponential and trigonometric functions. We will try to find $$x$$ by testing values or using numerical methods since algebraic isolation is complex. 3. **Step-by-step solution:** - Note that $$5 \times 90^\circ = 450^\circ$$. - Convert degrees to radians if needed for $$\tan(3x)$$, but since $$x$$ is in degrees, keep consistent. - Try integer values of $$x$$ from 0 to 450 and check the equation. 4. **Testing values:** - At $$x=2$$: $$24^2 = 576$$ $$10^2 = 100$$ $$14^2 = 196$$ $$\tan(6^\circ) \approx 0.1051$$ Left side: $$576 - 2 \times 0.1051 - 100 = 576 - 0.2102 - 100 = 475.7898$$ Right side: $$196$$ Left > Right. - At $$x=3$$: $$24^3 = 13824$$ $$10^3 = 1000$$ $$14^3 = 2744$$ $$\tan(9^\circ) \approx 0.1584$$ Left side: $$13824 - 2 \times 0.1584 - 1000 = 13824 - 0.3168 - 1000 = 12823.6832$$ Right side: $$2744$$ Left > Right. - At $$x=1$$: $$24^1 = 24$$ $$10^1 = 10$$ $$14^1 = 14$$ $$\tan(3^\circ) \approx 0.0524$$ Left side: $$24 - 2 \times 0.0524 - 10 = 24 - 0.1048 - 10 = 13.8952$$ Right side: $$14$$ Left < Right. - At $$x=1.1$$ (approximate): $$24^{1.1} \approx 31.5$$ $$10^{1.1} \approx 12.6$$ $$14^{1.1} \approx 16.3$$ $$\tan(3.3^\circ) \approx 0.0576$$ Left side: $$31.5 - 2 \times 0.0576 - 12.6 = 31.5 - 0.1152 - 12.6 = 18.7848$$ Right side: $$16.3$$ Left > Right. - At $$x=1.05$$: $$24^{1.05} \approx 27.5$$ $$10^{1.05} \approx 11.2$$ $$14^{1.05} \approx 15.1$$ $$\tan(3.15^\circ) \approx 0.0550$$ Left side: $$27.5 - 2 \times 0.0550 - 11.2 = 27.5 - 0.11 - 11.2 = 16.19$$ Right side: $$15.1$$ Left > Right. - At $$x=1$$ Left < Right, at $$x=1.05$$ Left > Right, so root is near $$x=1$$. 5. **Conclusion:** The value of $$x$$ that satisfies the equation is approximately $$1$$ to the nearest whole number.