Solve the inequality (x + 1)(x − 4) ≤ 1/4.

Step-by-step solution

1. 1

   Move everything to one side first. A product is only easy to read off when it is compared to zero, and here the right-hand side is $\tfrac14$, not $0$ — so the factored form on the left is of no immediate use. Expand and rearrange:

   $x^{2}-3x-4\le \frac14\quad\Longrightarrow\quad x^{2}-3x-\frac{17}{4}\le 0$

2. 2

   Clear the fraction. Multiplying by the positive number $4$ leaves the inequality direction unchanged:

   $4x^{2}-12x-17\le 0$

3. 3

   Find the roots with the quadratic formula.

   $x=\frac{12\pm\sqrt{144-4(4)(-17)}}{8}=\frac{12\pm\sqrt{144+272}}{8}=\frac{12\pm\sqrt{416}}{8}$

   Since $416=16\times 26$, $\sqrt{416}=4\sqrt{26}$, so

   $x=\frac{12\pm 4\sqrt{26}}{8}=\frac{3\pm\sqrt{26}}{2}$

4. 4

   Use the direction of opening. The coefficient of $x^2$ is $4>0$, so the parabola opens upward and is $\le 0$ between its roots (inclusive, because the inequality is not strict):

   $\boxed{\frac{3-\sqrt{26}}{2}\le x\le \frac{3+\sqrt{26}}{2}}$

5. 5

   Give the decimal form and check a point. Since $\sqrt{26}\approx 5.0990$, the interval is approximately $[-1.0495,\ 4.0495]$. Testing the midpoint $x=1.5$: $(2.5)(-2.5)=-6.25\le 0.25$ ✓. Testing $x=5$: $(6)(1)=6>0.25$ ✗, as expected outside the interval.

6. 6

   Note how close the endpoints are to the obvious guesses. The factored form tempts you to answer $-1\le x\le 4$, which is the solution of $(x+1)(x-4)\le 0$. The extra $\tfrac14$ pushes each endpoint outward by about $0.05$ — small, but the exact endpoints are irrational, not $-1$ and $4$.