Question Solve inequality $4|y-2|-3>17$ . If all real numbers are solutions, click "All reals". If no solution, click "No solution".

Studdy Solution

STEP 1

Assumptions
1. We are solving an inequality of the form $4|y-2|-3>17$ .
2. The variable $y$ is a real number.
3. The absolute value function $|y-2|$ represents the distance of $y-2$ from zero on the real number line.

STEP 2

First, we need to isolate the absolute value expression on one side of the inequality. To do this, we add 3 to both sides of the inequality.
$4|y-2|-3+3>17+3$

STEP 3

Simplify both sides of the inequality.
$4|y-2|>20$

STEP 4

Next, we divide both sides of the inequality by 4 to solve for the absolute value expression.
$\frac{4|y-2|}{4}>\frac{20}{4}$

STEP 5

Simplify both sides of the inequality.
$|y-2|>5$

STEP 6

The inequality $|y-2|>5$ means that the distance of $y-2$ from zero is greater than 5. This leads to two cases:
1. $y-2$ is greater than 5.
2. $y-2$ is less than -5.
We will solve these two cases separately.

STEP 7

For the first case, where $y-2$ is greater than 5, we set up the inequality:
$y-2>5$

STEP 8

Add 2 to both sides of the inequality to solve for $y$ .
$y-2+2>5+2$

STEP 9

Simplify the inequality.
$y>7$

STEP 10

For the second case, where $y-2$ is less than -5, we set up the inequality:
$y-2<-5$

STEP 11

Add 2 to both sides of the inequality to solve for $y$ .
$y-2+2<-5+2$

STEP 12

Simplify the inequality.
$y<-3$

STEP 13

Combine the solutions from both cases. The solution to the inequality $|y-2|>5$ is that $y$ is either greater than 7 or less than -3.
The solution to the inequality is:
$y>7 \quad \text{or} \quad y<-3$

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