confused on two inequality sided question

**GaDoomZ** · September 25th 2011, 12:24 PM

4x<2x+1<//<3x+2

I simplify it down to 2x<1<//<1x+2

but I don't know where to go from there

**GaDoomZ** · September 25th 2011, 12:30 PM

the second symbol is a less than or equal to sign I am posting from a tablet so didn't have the option

**Siron** · September 25th 2011, 12:40 PM

The intequality is $4x<2x+1\leq 3x+2$ , a possible way to solve this is by considering three cases:
(1) $4x<2x+1$
(2) $4x\leq3x+2$
(3) $2x+1\leq3x+2$

By solving this three inequality's and summarizing the solutions you will get the solutions for the original inequality.

**Archie Meade** · September 25th 2011, 01:45 PM

Originally Posted by GaDoomZ

4x<2x+1<//<3x+2

I simplify it down to 2x<1<//<1x+2

but I don't know where to go from there

You could subtract $2x+1$ from all three terms.

$4x-(2x+1)<2x+1-(2x+1)\le\ 3x+2-(2x+1)$

$2x-1<0\le\ x+1$

$2x-1<0$ is saying $2x<1\Rightarrow\ x<\frac{1}{2}$

$0\le\ x+1$ is saying $x\ge\ -1$

Then

$-1\le\ x<\frac{1}{2}$

**Plato** · September 25th 2011, 01:58 PM

Originally Posted by Siron

The intequality is $4x<2x+1\leq 3x+2$ , a possible way to solve this is by considering three cases:
(1) $4x<2x+1$
(2) $4x\leq3x+2$
(3) $2x+1\leq3x+2$

I like this approach.
Solution sets:
$\begin{gathered} (1)\;A = \left( { - \infty ,\frac{1}{2}} \right) \hfill \\ (2)\;B = \left( { - \infty ,2} \right] \hfill \\ (3)\;C = \left[ { - 1,\infty } \right) \hfill \\ \end{gathered}$

So $A \cap B \cap C = \left[ { - 1,\frac{1}{2}} \right)$

**Soroban** · September 25th 2011, 02:00 PM

Hello, GaDoomZ!

$4x \:<\:2x+1 \:\le\: 3x+2$

Separate the inequalities:

. . $4x \:<\:2x+1 \quad\Rightarrow\quad 2x \:<\:1 \quad\Rightarrow\quad x \:<\:\tfrac{1}{2}$

. . $2x+1\:\le\:3x+2 \quad\Rightarrow\quad -x \:\le\:1 \quad\Rightarrow\quad x \:\ge\:-1$

Therefore: . $-1\:\le\:x\:<\:\tfrac{1}{2}$

. . $\begin{array}{cccccc}---\!\!\!\!\! & \bullet\!\!\!\!\! & ===\!\!\!\!\! & \circ\!\!\!\!\! & ---\!\!\!\!\! \\ & \;\:\text{-}1 && \quad\frac{1}{2} \end{array}$

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