Solve the inequality 55(x^2 - 25) > 35(24^2 - 25). Give the exact bounds and their decimal values.

Step-by-step solution

1. 1

   Evaluate the constant side first. The right-hand side contains no unknown, so reduce it to a single number before touching $x$:

   $24^{2}=576,\qquad 576-25=551,\qquad 35\times551=19285.$

   The inequality becomes $55\left(x^{2}-25\right)>19285$. Doing this first keeps the later arithmetic to one division rather than several.

2. 2

   Divide by the positive coefficient. Since $55>0$, dividing both sides preserves the direction:

   $x^{2}-25>\frac{19285}{55}=\frac{3857}{11}.$

   The fraction reduces by $5$ ($19285=5\times3857$, $55=5\times11$) and does not reduce further, since $3857=7\times19\times29$ shares no factor with $11$. Keeping it exact avoids rounding the final bound.

3. 3

   Isolate $x^{2}$. Adding $25=\tfrac{275}{11}$ to both sides:

   $x^{2}>\frac{3857}{11}+\frac{275}{11}=\frac{4132}{11}\approx375.636.$

4. 4

   Solve the squared inequality with both branches. For a positive constant $c$, the statement $x^{2}>c$ holds outside the interval $\left[-\sqrt c,\sqrt c\right]$:

   $x<-\sqrt{\frac{4132}{11}}\quad\text{or}\quad x>\sqrt{\frac{4132}{11}}.$

   Writing only $x>\sqrt c$ loses the entire negative branch — the most common mistake on this type.

5. 5

   Simplify the surd and check. Since $4132=4\times1033$, rationalising the denominator gives

   $\sqrt{\frac{4132}{11}}=\frac{2\sqrt{1033}}{\sqrt{11}}=\frac{2\sqrt{11363}}{11}\approx19.3813.$

   Testing: at $x=19.5$, $55(380.25-25)=19538.75>19285$ ✓; at $x=19.3$, $55(372.49-25)=19111.95$, which fails ✓. By symmetry $x=-19.5$ passes and $x=-19.3$ fails ✓.