Question13. Solve the inequality. $3|1-x|-1<5$

Studdy Solution

STEP 1

What is this asking? We need to find all the values of $x$ that make the inequality $3|1-x|-1<5$ true! Watch out! Don't forget how absolute values work!
They make things positive, so we need to consider both positive and negative cases.

STEP 2

1. Isolate the absolute value
2. Consider the two cases
3. Solve for x in each case
4. Combine the solutions

STEP 3

First, let's isolate that absolute value.
We can do this by adding $1$ to both sides of the inequality: $3|1-x|-1+1 < 5+1$ $3|1-x| < 6$

STEP 4

Now, let's divide both sides by $3$ : $\frac{3|1-x|}{3} < \frac{6}{3}$ $|1-x| < 2$ Great! Now we have the absolute value all by itself!

STEP 5

If $1-x$ is positive or zero, then $|1-x| = 1-x$ .
So, we have: $1-x < 2$

STEP 6

If $1-x$ is negative, then $|1-x| = -(1-x) = x-1$ .
So, we have: $x-1 < 2$

STEP 7

$1-x < 2$ Subtract $1$ from both sides: $1-x-1 < 2-1$ $-x < 1$ Multiply both sides by $-1$ and flip the inequality sign: $x > -1$

STEP 8

$x-1 < 2$ Add $1$ to both sides: $x-1+1 < 2+1$ $x < 3$

STEP 9

We found that $x > -1$ and $x < 3$ .
This means $x$ has to be greater than $-1$ and less than $3$ .
We can write this as a single inequality: $-1 < x < 3$

STEP 10

All values of $x$ between $-1$ and $3$ (not including $-1$ and $3$ ) satisfy the inequality.
So our final answer is $-1 < x < 3$ .

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