Collection of Maths Problems

Inequalities with fractions

Task number: 2758

• Variant 1

  \( \frac{x+1}{x-2}\ge 0 \)

• Resolution

  The expresssion is undefined for \(x=2\).

  \(x<2: \longrightarrow x+1\le 0 \longrightarrow x\le-1 \longrightarrow x\in (-\infty,-1\rangle\)

  \(x>2: \longrightarrow x+1\ge 0 \longrightarrow x\ge-1 \longrightarrow x\in (2,\infty)\)

• Result

  The solution is \(x\in (-\infty,-1\rangle \cup (2,\infty)\).

• Variant 2

  \( \frac{x-2}{2x-8}\ge 1 \)

• Resolution

  The expression is undefined for \(2x-8=0\) or \(x=4\).

  • \(x< 4: x-2\le 2x-8 \longrightarrow x\ge 6 \longrightarrow \) has no solution
  • \(x> 4: x-2\ge 2x-8 \longrightarrow x\le 6 \longrightarrow x\in (4{,}6\rangle\)

• Result

  The solution is \(x\in (4{,}6\rangle\).

• Variant 3

  \( \frac{x+3}{x-1}\ge \frac{x+1}{x-5} \)

• Resolution

  The expression is undefined for \(x=1\) or \(x=5\).

  • \(x< 1 : (x+3)(x-5)\ge x^2-1 \longrightarrow x\le 7 \longrightarrow x\in(-\infty,1)\)
  • \(1< x<5 : (x+3)(x-5)\le x^2-1 \longrightarrow x\ge 7 \longrightarrow \)has no solution
  • \(x>5 : (x+3)(x-5)\ge x^2-1 \longrightarrow x\le 7 \longrightarrow x\in(5{,}7\rangle\)

• Result

  The solution is \(x\in(-\infty,1) \cup (5{,}7\rangle\).

• Variant 4

  \( \frac1{x+2}<\frac{x}{x-1} \)

• Hint

  The expression is undefined for \(x=-2\) or \(x=1\).

  • \(x<-2: x-1< x(x+2) \longrightarrow x^2+x+1 > 0 \longrightarrow x_{1{,}2}=\frac{-1\pm \sqrt{1-4}}{2} \longrightarrow x\in(-\infty,-2)\)
  • \(-2< x<1: x-1> x(x+2) \longrightarrow x^2+x+1 < 0 \longrightarrow\) has no solution
  • \(x>1: x-1< x(x+2) \longrightarrow x^2+x+1 > 0 \longrightarrow x\in (1,\infty)\)

• Resolution

  The solution is \(x\in(-\infty,-2)\cup(1,\infty)\).