A chained inequality means an intersection

Without a graphing calculator solve x2/(x−1)≤4x≤8x^2/(x-1)\le4x\le8.

[2011/MJC/Prelim/I/6a]

Solution

Q

Compound inequalities

Without a graphing calculator, solve x2x−1≤4x≤8\dfrac{x^2}{x-1}\le4x\le8.

Explore this idea

Hint

  1. Split into two inequalities: x2x−1≤4x\frac{x^2}{x-1}\le4x and 4x≤84x\le8.

  2. Solve the first with a sign chart (remember x≠1x\ne1), the second directly.

  3. Intersect the solution sets.

Answer · Q

0≤x<10\le x<1 or 4/3≤x≤24/3\le x\le2.

Explore the idea

Where is x(4 − 3x)/(x − 1) ≤ 0?

The shaded strips solve the first inequality. The second, x ≤ 2, then cuts off everything to the right of 2.

x(4−3x)x−1≤0\frac{x(4-3x)}{x-1}\le0

Choose an interval to test it. Shading marks the solution intervals.

Intervalxx − 4/3x − 1product
negativenegativenegativepositive
positivenegativenegativenegative
positivenegativepositivepositive
positivepositivepositivenegative
Test value of xx = 1.50
Try an example

(x)+(x − 4/3)+(x − 1)+⇒ product negative

Is it ≤ 0?
Yes
x is in the solution set

The shaded strips are where the expression satisfies the inequality; they match the green-tinted columns of the chart.

  • Expression and where it is ≤ 0
  • Your test value
Key idea

Each link has its own solution set. The chain's solution is where they overlap: here, the sign-chart answer with everything above 2 removed.

Ideas to try
  1. The chart says x=3x=3 solves the first link. Does it solve the whole chain?
  2. Test x=43x=\frac43: is the end-point included?