Question Find solutions to $|5x-2| = 12$ . Options: a. $\frac{14}{5}$ , b. -2, c. 2, d. $-\frac{14}{5}$ .

Studdy Solution

STEP 1

Assumptions
1. The function is an absolute value function of the form $g(x) = |ax + b| = c$ .
2. We are given $g(x) = |-5x + 2| = 12$ .
3. We need to find all $x$ that satisfy the equation.

STEP 2

The absolute value function $|ax + b| = c$ has two possible cases because the absolute value of a number is always non-negative. Therefore, the expression inside the absolute value can be either positive or equal to $c$ or negative and its absolute value equal to $c$ .

STEP 3

We will solve the equation for both cases. For the first case, we assume the expression inside the absolute value is non-negative:
$-5x + 2 = 12$

STEP 4

Subtract 2 from both sides of the equation to isolate the term with $x$ :
$-5x = 12 - 2$

STEP 5

Calculate the right side of the equation:
$-5x = 10$

STEP 6

Divide both sides by $-5$ to solve for $x$ :
$x = \frac{10}{-5}$

STEP 7

Calculate the value of $x$ :
$x = -2$

STEP 8

For the second case, we assume the expression inside the absolute value is negative and its absolute value equals $c$ :
$-(-5x + 2) = 12$

STEP 9

Simplify the equation by distributing the negative sign:
$5x - 2 = 12$

STEP 10

Add 2 to both sides of the equation to isolate the term with $x$ :
$5x = 12 + 2$

STEP 11

Calculate the right side of the equation:
$5x = 14$

STEP 12

Divide both sides by 5 to solve for $x$ :
$x = \frac{14}{5}$

STEP 13

Now we have found both possible solutions for $x$ . The solutions are:
$x = -2 \quad \text{and} \quad x = \frac{14}{5}$

STEP 14

We compare the solutions with the given options:
a. $\frac{14}{5}$ (Correct) b. -2 (Correct) c. 2 (Incorrect) d. $-\frac{14}{5}$ (Incorrect)

STEP 15

The correct answers are:
a. $\frac{14}{5}$ b. -2

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Question Find solutions to $|5x-2| = 12$ . Options: a. $\frac{14}{5}$ , b. -2, c. 2, d. $-\frac{14}{5}$ .

Studdy Solution

STEP 1

Assumptions
1. The function is an absolute value function of the form $g(x) = |ax + b| = c$ .
2. We are given $g(x) = |-5x + 2| = 12$ .
3. We need to find all $x$ that satisfy the equation.

STEP 2

The absolute value function $|ax + b| = c$ has two possible cases because the absolute value of a number is always non-negative. Therefore, the expression inside the absolute value can be either positive or equal to $c$ or negative and its absolute value equal to $c$ .

STEP 3

We will solve the equation for both cases. For the first case, we assume the expression inside the absolute value is non-negative:
$-5x + 2 = 12$

STEP 4

Subtract 2 from both sides of the equation to isolate the term with $x$ :
$-5x = 12 - 2$

STEP 5

Calculate the right side of the equation:
$-5x = 10$

STEP 6

Divide both sides by $-5$ to solve for $x$ :
$x = \frac{10}{-5}$

STEP 7

Calculate the value of $x$ :
$x = -2$

STEP 8

For the second case, we assume the expression inside the absolute value is negative and its absolute value equals $c$ :
$-(-5x + 2) = 12$

STEP 9

Simplify the equation by distributing the negative sign:
$5x - 2 = 12$

STEP 10

Add 2 to both sides of the equation to isolate the term with $x$ :
$5x = 12 + 2$

STEP 11

Calculate the right side of the equation:
$5x = 14$

STEP 12

Divide both sides by 5 to solve for $x$ :
$x = \frac{14}{5}$

STEP 13

Now we have found both possible solutions for $x$ . The solutions are:
$x = -2 \quad \text{and} \quad x = \frac{14}{5}$

STEP 14

We compare the solutions with the given options:
a. $\frac{14}{5}$ (Correct) b. -2 (Correct) c. 2 (Incorrect) d. $-\frac{14}{5}$ (Incorrect)

STEP 15

The correct answers are:
a. $\frac{14}{5}$ b. -2

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Question Find solutions to ∣5x−2∣=12|5x-2| = 12∣5x−2∣=12. Options: a. 145\frac{14}{5}514​, b. -2, c. 2, d. −145-\frac{14}{5}−514​.

Studdy Solution

STEP 1

Assumptions1. The function is an absolute value function of the form g(x)=∣ax+b∣=cg(x) = |ax + b| = cg(x)=∣ax+b∣=c.2. We are given g(x)=∣−5x+2∣=12g(x) = |-5x + 2| = 12g(x)=∣−5x+2∣=12.3. We need to find all xxx that satisfy the equation.

STEP 2

The absolute value function ∣ax+b∣=c|ax + b| = c∣ax+b∣=c has two possible cases because the absolute value of a number is always non-negative. Therefore, the expression inside the absolute value can be either positive or equal to ccc or negative and its absolute value equal to ccc.

STEP 3

We will solve the equation for both cases. For the first case, we assume the expression inside the absolute value is non-negative:−5x+2=12-5x + 2 = 12−5x+2=12

STEP 4

Subtract 2 from both sides of the equation to isolate the term with xxx:−5x=12−2-5x = 12 - 2−5x=12−2

STEP 5

Calculate the right side of the equation:−5x=10-5x = 10−5x=10

STEP 6

Divide both sides by −5-5−5 to solve for xxx:x=10−5x = \frac{10}{-5}x=−510​

STEP 7

Calculate the value of xxx:x=−2x = -2x=−2

STEP 8

For the second case, we assume the expression inside the absolute value is negative and its absolute value equals ccc:−(−5x+2)=12-(-5x + 2) = 12−(−5x+2)=12

STEP 9

Simplify the equation by distributing the negative sign:5x−2=125x - 2 = 125x−2=12

STEP 10

Add 2 to both sides of the equation to isolate the term with xxx:5x=12+25x = 12 + 25x=12+2

STEP 11

Calculate the right side of the equation:5x=145x = 145x=14

STEP 12

Divide both sides by 5 to solve for xxx:x=145x = \frac{14}{5}x=514​

STEP 13

Now we have found both possible solutions for xxx. The solutions are:x=−2andx=145x = -2 \quad \text{and} \quad x = \frac{14}{5}x=−2andx=514​

STEP 14

We compare the solutions with the given options:a. 145\frac{14}{5}514​ (Correct) b. -2 (Correct) c. 2 (Incorrect) d. −145-\frac{14}{5}−514​ (Incorrect)

STEP 15

The correct answers are:a. 145\frac{14}{5}514​ b. -2

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Question Find solutions to $|5x-2| = 12$ . Options: a. $\frac{14}{5}$ , b. -2, c. 2, d. $-\frac{14}{5}$ .

Assumptions
1. The function is an absolute value function of the form $g(x) = |ax + b| = c$ .
2. We are given $g(x) = |-5x + 2| = 12$ .
3. We need to find all $x$ that satisfy the equation.

The absolute value function $|ax + b| = c$ has two possible cases because the absolute value of a number is always non-negative. Therefore, the expression inside the absolute value can be either positive or equal to $c$ or negative and its absolute value equal to $c$ .

We will solve the equation for both cases. For the first case, we assume the expression inside the absolute value is non-negative:
$-5x + 2 = 12$

Subtract 2 from both sides of the equation to isolate the term with $x$ :
$-5x = 12 - 2$

Calculate the right side of the equation:
$-5x = 10$

Divide both sides by $-5$ to solve for $x$ :
$x = \frac{10}{-5}$

Calculate the value of $x$ :
$x = -2$

For the second case, we assume the expression inside the absolute value is negative and its absolute value equals $c$ :
$-(-5x + 2) = 12$

Simplify the equation by distributing the negative sign:
$5x - 2 = 12$

Add 2 to both sides of the equation to isolate the term with $x$ :
$5x = 12 + 2$

Calculate the right side of the equation:
$5x = 14$

Divide both sides by 5 to solve for $x$ :
$x = \frac{14}{5}$

Now we have found both possible solutions for $x$ . The solutions are:
$x = -2 \quad \text{and} \quad x = \frac{14}{5}$

We compare the solutions with the given options:
a. $\frac{14}{5}$ (Correct) b. -2 (Correct) c. 2 (Incorrect) d. $-\frac{14}{5}$ (Incorrect)

The correct answers are:
a. $\frac{14}{5}$ b. -2