Solved on Dec 05, 2023

Solve the absolute value equation $|3x+5| = 1$ for the value of $x$ .

STEP 1

Assumptions
1. We are given the absolute value equation $|3x + 5| = 1$ .
2. We need to find all values of $x$ that satisfy the equation.

STEP 2

Understand that the absolute value of a number is its distance from zero on the number line, which means it is always non-negative. The absolute value equation $|3x + 5| = 1$ tells us that the expression $3x + 5$ is 1 unit away from zero on the number line.

STEP 3

Recognize that if $|A| = B$ where $B \geq 0$ , then $A = B$ or $A = -B$ . This is because the absolute value of $A$ could be $B$ if $A$ is already non-negative, or it could be $B$ if $A$ is negative and its absolute value is $B$ .

STEP 4

Apply the principle from STEP_3 to the given equation $|3x + 5| = 1$ . This gives us two possible equations:
1. $3x + 5 = 1$
2. $3x + 5 = -1$

STEP 5

Solve the first equation $3x + 5 = 1$ .
Subtract 5 from both sides of the equation to isolate the term with $x$ .
$3x + 5 - 5 = 1 - 5$

STEP 6

Simplify the equation from STEP_5.
$3x = -4$

STEP 7

Divide both sides of the equation by 3 to solve for $x$ .
$\frac{3x}{3} = \frac{-4}{3}$

STEP 8

Simplify the equation from STEP_7 to find the first solution for $x$ .
$x = -\frac{4}{3}$

STEP 9

Now, solve the second equation $3x + 5 = -1$ .
Subtract 5 from both sides of the equation to isolate the term with $x$ .
$3x + 5 - 5 = -1 - 5$

STEP 10

Simplify the equation from STEP_9.
$3x = -6$

STEP 11

Divide both sides of the equation by 3 to solve for $x$ .
$\frac{3x}{3} = \frac{-6}{3}$

STEP 12

Simplify the equation from STEP_11 to find the second solution for $x$ .
$x = -2$
The solutions to the equation $|3x + 5| = 1$ are $x = -\frac{4}{3}$ and $x = -2$ .

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Solve the absolute value equation $|3x+5| = 1$ for the value of $x$ .

STEP 1

Assumptions
1. We are given the absolute value equation $|3x + 5| = 1$ .
2. We need to find all values of $x$ that satisfy the equation.

STEP 2

Understand that the absolute value of a number is its distance from zero on the number line, which means it is always non-negative. The absolute value equation $|3x + 5| = 1$ tells us that the expression $3x + 5$ is 1 unit away from zero on the number line.

STEP 3

Recognize that if $|A| = B$ where $B \geq 0$ , then $A = B$ or $A = -B$ . This is because the absolute value of $A$ could be $B$ if $A$ is already non-negative, or it could be $B$ if $A$ is negative and its absolute value is $B$ .

STEP 4

Apply the principle from STEP_3 to the given equation $|3x + 5| = 1$ . This gives us two possible equations:
1. $3x + 5 = 1$
2. $3x + 5 = -1$

STEP 5

Solve the first equation $3x + 5 = 1$ .
Subtract 5 from both sides of the equation to isolate the term with $x$ .
$3x + 5 - 5 = 1 - 5$

STEP 6

Simplify the equation from STEP_5.
$3x = -4$

STEP 7

Divide both sides of the equation by 3 to solve for $x$ .
$\frac{3x}{3} = \frac{-4}{3}$

STEP 8

Simplify the equation from STEP_7 to find the first solution for $x$ .
$x = -\frac{4}{3}$

STEP 9

Now, solve the second equation $3x + 5 = -1$ .
Subtract 5 from both sides of the equation to isolate the term with $x$ .
$3x + 5 - 5 = -1 - 5$

STEP 10

Simplify the equation from STEP_9.
$3x = -6$

STEP 11

Divide both sides of the equation by 3 to solve for $x$ .
$\frac{3x}{3} = \frac{-6}{3}$

STEP 12

Simplify the equation from STEP_11 to find the second solution for $x$ .
$x = -2$
The solutions to the equation $|3x + 5| = 1$ are $x = -\frac{4}{3}$ and $x = -2$ .

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Solve the absolute value equation ∣3x+5∣=1|3x+5| = 1∣3x+5∣=1 for the value of xxx.

STEP 1

Assumptions1. We are given the absolute value equation ∣3x+5∣=1|3x + 5| = 1∣3x+5∣=1.2. We need to find all values of xxx that satisfy the equation.

STEP 2

Understand that the absolute value of a number is its distance from zero on the number line, which means it is always non-negative. The absolute value equation ∣3x+5∣=1|3x + 5| = 1∣3x+5∣=1 tells us that the expression 3x+53x + 53x+5 is 1 unit away from zero on the number line.

STEP 3

Recognize that if ∣A∣=B|A| = B∣A∣=B where B≥0B \geq 0B≥0, then A=BA = BA=B or A=−BA = -BA=−B. This is because the absolute value of AAA could be BBB if AAA is already non-negative, or it could be BBB if AAA is negative and its absolute value is BBB.

STEP 4

Apply the principle from STEP_3 to the given equation ∣3x+5∣=1|3x + 5| = 1∣3x+5∣=1. This gives us two possible equations:1. 3x+5=13x + 5 = 13x+5=12. 3x+5=−13x + 5 = -13x+5=−1

STEP 5

Solve the first equation 3x+5=13x + 5 = 13x+5=1.Subtract 5 from both sides of the equation to isolate the term with xxx.3x+5−5=1−53x + 5 - 5 = 1 - 53x+5−5=1−5

STEP 6

Simplify the equation from STEP_5.3x=−43x = -43x=−4

STEP 7

Divide both sides of the equation by 3 to solve for xxx.3x3=−43\frac{3x}{3} = \frac{-4}{3}33x​=3−4​

STEP 8

Simplify the equation from STEP_7 to find the first solution for xxx.x=−43x = -\frac{4}{3}x=−34​

STEP 9

Now, solve the second equation 3x+5=−13x + 5 = -13x+5=−1.Subtract 5 from both sides of the equation to isolate the term with xxx.3x+5−5=−1−53x + 5 - 5 = -1 - 53x+5−5=−1−5

STEP 10

Simplify the equation from STEP_9.3x=−63x = -63x=−6

STEP 11

Divide both sides of the equation by 3 to solve for xxx.3x3=−63\frac{3x}{3} = \frac{-6}{3}33x​=3−6​

STEP 12

Simplify the equation from STEP_11 to find the second solution for xxx.x=−2x = -2x=−2The solutions to the equation ∣3x+5∣=1|3x + 5| = 1∣3x+5∣=1 are x=−43x = -\frac{4}{3}x=−34​ and x=−2x = -2x=−2.

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Solve the absolute value equation $|3x+5| = 1$ for the value of $x$ .

Assumptions
1. We are given the absolute value equation $|3x + 5| = 1$ .
2. We need to find all values of $x$ that satisfy the equation.

Understand that the absolute value of a number is its distance from zero on the number line, which means it is always non-negative. The absolute value equation $|3x + 5| = 1$ tells us that the expression $3x + 5$ is 1 unit away from zero on the number line.

Recognize that if $|A| = B$ where $B \geq 0$ , then $A = B$ or $A = -B$ . This is because the absolute value of $A$ could be $B$ if $A$ is already non-negative, or it could be $B$ if $A$ is negative and its absolute value is $B$ .

Apply the principle from STEP_3 to the given equation $|3x + 5| = 1$ . This gives us two possible equations:
1. $3x + 5 = 1$
2. $3x + 5 = -1$

Solve the first equation $3x + 5 = 1$ .
Subtract 5 from both sides of the equation to isolate the term with $x$ .
$3x + 5 - 5 = 1 - 5$

Simplify the equation from STEP_5.
$3x = -4$

Divide both sides of the equation by 3 to solve for $x$ .
$\frac{3x}{3} = \frac{-4}{3}$

Simplify the equation from STEP_7 to find the first solution for $x$ .
$x = -\frac{4}{3}$

Now, solve the second equation $3x + 5 = -1$ .
Subtract 5 from both sides of the equation to isolate the term with $x$ .
$3x + 5 - 5 = -1 - 5$

Simplify the equation from STEP_9.
$3x = -6$

Divide both sides of the equation by 3 to solve for $x$ .
$\frac{3x}{3} = \frac{-6}{3}$

Simplify the equation from STEP_11 to find the second solution for $x$ .
$x = -2$
The solutions to the equation $|3x + 5| = 1$ are $x = -\frac{4}{3}$ and $x = -2$ .