QuestionSolve the equation: $|3y + 6| = 13$ . What is the solution set? A. (Type your answer) B. The solution set is $\varnothing$ .

Studdy Solution

STEP 1

Assumptions1. The absolute value of a number is its distance from zero on the number line, regardless of direction. Therefore, it is always non-negative. . The equation to solve is $|3y+6|=13$ .

STEP 2

An absolute value equation can be rewritten as two separate equations one for the positive case and one for the negative case. This is because if $|a|=b$ , then $a=b$ or $a=-b$ .
$y+6=13$ or $y+6=-13$

STEP 3

Now, we solve the first equation $3y+6=13$ for $y$ . We start by subtracting6 from both sides of the equation.
$3y+6-6=13-6$

STEP 4

implify the equation to isolate $3y$ on one side.
$3y=7$

STEP 5

Divide both sides of the equation by3 to solve for $y$ .
$y=\frac{7}{3}$

STEP 6

Now, we solve the second equation $3y+6=-13$ for $y$ . We start by subtracting6 from both sides of the equation.
$3y+6-6=-13-6$

STEP 7

implify the equation to isolate $3y$ on one side.
$3y=-19$

STEP 8

Divide both sides of the equation by3 to solve for $y$ .
$y=-\frac{19}{3}$

STEP 9

The solution set to the absolute value equation $|3y+6|=13$ is the set of all $y$ that satisfy either $3y+6=13$ or $3y+6=-13$ . Therefore, the solution set is $\left\{\frac{7}{3}, -\frac{19}{3}\right\}$ .
The solution set is $\left\{\frac{7}{3}, -\frac{19}{3}\right\}$ .

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QuestionSolve the equation: $|3y + 6| = 13$ . What is the solution set? A. (Type your answer) B. The solution set is $\varnothing$ .

Studdy Solution

STEP 1

Assumptions1. The absolute value of a number is its distance from zero on the number line, regardless of direction. Therefore, it is always non-negative. . The equation to solve is $|3y+6|=13$ .

STEP 2

An absolute value equation can be rewritten as two separate equations one for the positive case and one for the negative case. This is because if $|a|=b$ , then $a=b$ or $a=-b$ .
$y+6=13$ or $y+6=-13$

STEP 3

Now, we solve the first equation $3y+6=13$ for $y$ . We start by subtracting6 from both sides of the equation.
$3y+6-6=13-6$

STEP 4

implify the equation to isolate $3y$ on one side.
$3y=7$

STEP 5

Divide both sides of the equation by3 to solve for $y$ .
$y=\frac{7}{3}$

STEP 6

Now, we solve the second equation $3y+6=-13$ for $y$ . We start by subtracting6 from both sides of the equation.
$3y+6-6=-13-6$

STEP 7

implify the equation to isolate $3y$ on one side.
$3y=-19$

STEP 8

Divide both sides of the equation by3 to solve for $y$ .
$y=-\frac{19}{3}$

STEP 9

The solution set to the absolute value equation $|3y+6|=13$ is the set of all $y$ that satisfy either $3y+6=13$ or $3y+6=-13$ . Therefore, the solution set is $\left\{\frac{7}{3}, -\frac{19}{3}\right\}$ .
The solution set is $\left\{\frac{7}{3}, -\frac{19}{3}\right\}$ .

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QuestionSolve the equation: ∣3y+6∣=13|3y + 6| = 13∣3y+6∣=13. What is the solution set? A. (Type your answer) B. The solution set is ∅\varnothing∅.

Studdy Solution

STEP 1

Assumptions1. The absolute value of a number is its distance from zero on the number line, regardless of direction. Therefore, it is always non-negative. . The equation to solve is ∣3y+6∣=13|3y+6|=13∣3y+6∣=13.

STEP 2

An absolute value equation can be rewritten as two separate equations one for the positive case and one for the negative case. This is because if ∣a∣=b|a|=b∣a∣=b, then a=ba=ba=b or a=−ba=-ba=−b.y+6=13y+6=13y+6=13ory+6=−13y+6=-13y+6=−13

STEP 3

Now, we solve the first equation 3y+6=133y+6=133y+6=13 for yyy. We start by subtracting6 from both sides of the equation.3y+6−6=13−63y+6-6=13-63y+6−6=13−6

STEP 4

implify the equation to isolate 3y3y3y on one side.3y=73y=73y=7

STEP 5

Divide both sides of the equation by3 to solve for yyy.y=73y=\frac{7}{3}y=37​

STEP 6

Now, we solve the second equation 3y+6=−133y+6=-133y+6=−13 for yyy. We start by subtracting6 from both sides of the equation.3y+6−6=−13−63y+6-6=-13-63y+6−6=−13−6

STEP 7

implify the equation to isolate 3y3y3y on one side.3y=−193y=-193y=−19

STEP 8

Divide both sides of the equation by3 to solve for yyy.y=−193y=-\frac{19}{3}y=−319​

STEP 9

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QuestionSolve the equation: $|3y + 6| = 13$ . What is the solution set? A. (Type your answer) B. The solution set is $\varnothing$ .

Assumptions1. The absolute value of a number is its distance from zero on the number line, regardless of direction. Therefore, it is always non-negative. . The equation to solve is $|3y+6|=13$ .

An absolute value equation can be rewritten as two separate equations one for the positive case and one for the negative case. This is because if $|a|=b$ , then $a=b$ or $a=-b$ .
$y+6=13$ or $y+6=-13$

Now, we solve the first equation $3y+6=13$ for $y$ . We start by subtracting6 from both sides of the equation.
$3y+6-6=13-6$

implify the equation to isolate $3y$ on one side.
$3y=7$

Divide both sides of the equation by3 to solve for $y$ .
$y=\frac{7}{3}$

Now, we solve the second equation $3y+6=-13$ for $y$ . We start by subtracting6 from both sides of the equation.
$3y+6-6=-13-6$

implify the equation to isolate $3y$ on one side.
$3y=-19$

Divide both sides of the equation by3 to solve for $y$ .
$y=-\frac{19}{3}$