QuestionSolve the inequality $|x-1|-2x>1$ .

Studdy Solution

STEP 1

Assumptions1. The problem is to solve the inequality $|x-1|- > x$ . . The absolute value function $|x-1|$ can be defined as $x-1$ if $x \geq1$ and as $-(x-1)$ if $x <1$ .

STEP 2

We will solve the inequality in two parts, one for $x \geq1$ and another for $x <1$ .
First, let's solve for $x \geq1$ . In this case, $|x-1|$ can be replaced by $x-1$ .
$x -1 -2 > x$

STEP 3

implify the inequality by subtracting $x$ from both sides.
$-1 -2 >0$

STEP 4

Further simplify the inequality.
$-3 >0$ This inequality is not true, so there are no solutions for $x \geq1$ .

STEP 5

Now, let's solve for $x <1$ . In this case, $|x-1|$ can be replaced by $-(x-1)$ .
$-(x -1) -2 > x$

STEP 6

implify the inequality by distributing the negative sign.
$-x +1 -2 > x$

STEP 7

Further simplify the inequality by combining like terms.
$-x -1 > x$

STEP 8

Add $x$ to both sides of the inequality to isolate $x$ on one side.
$-1 >2x$

STEP 9

Divide both sides of the inequality by $2$ to solve for $x$ .
$-.5 > x$ This inequality is true for $x < -.5$ .

STEP 10

Combine the solutions from both parts. Since there are no solutions for $x \geq$ and the solution for $x <$ is $x < -0.5$ , the solution to the inequality $|x-|-2 > x$ is $x < -0.5$ .
So, the solution to the inequality is $x < -0.5$ .

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