Question

question 6
solve the equation $10|2x - 4| - 3 = 6$.
(fill each blank with an integer where the smaller integer is on the first blank.)
$x = \square$ or $x = \square$

Explanation:

Step1: Isolate the absolute value term

$10|2x - 4| = 6 + 3$
$10|2x - 4| = 9$
$|2x - 4| = \frac{9}{10}$

Step2: Split into two linear equations

$2x - 4 = \frac{9}{10}$ or $2x - 4 = -\frac{9}{10}$

Step3: Solve first linear equation

$2x = \frac{9}{10} + 4$
$2x = \frac{9}{10} + \frac{40}{10} = \frac{49}{10}$
$x = \frac{49}{20} = \frac{9}{2}$ (corrected calculation: $4 = \frac{40}{10}$, so $\frac{9}{10}+\frac{40}{10}=\frac{49}{10}$ → $x=\frac{49}{20}$ was error, correct: $10|2x-4|=9$ → $|2x-4|=0.9$
First case: $2x-4=0.9$ → $2x=4.9$ → $x=2.45=\frac{49}{20}$
Second case: $2x-4=-0.9$ → $2x=3.1$ → $x=1.55=\frac{31}{20}$
Wait, original equation: $10|2x-4|-3=6$ → $10|2x-4|=9$ → $|2x-4|=0.9$
Step1: Add 3 to both sides
$10|2x-4|=6+3=9$
Step2: Divide by 10
$|2x-4|=\frac{9}{10}$
Step3: Solve $2x-4=\frac{9}{10}$
$2x=4+\frac{9}{10}=\frac{40}{10}+\frac{9}{10}=\frac{49}{10}$
$x=\frac{49}{20}=2.45$
Step4: Solve $2x-4=-\frac{9}{10}$
$2x=4-\frac{9}{10}=\frac{40}{10}-\frac{9}{10}=\frac{31}{10}$
$x=\frac{31}{20}=1.55$
Correct final answer: $x=\frac{31}{20}$ or $x=\frac{49}{20}$

Step1: Isolate absolute value

$10|2x-4|=6+3=9$

Step2: Divide by 10

$|2x-4|=\frac{9}{10}$

Step3: Solve positive case

$2x-4=\frac{9}{10}$
$2x=\frac{49}{10}$
$x=\frac{49}{20}$

Step4: Solve negative case

$2x-4=-\frac{9}{10}$
$2x=\frac{31}{10}$
$x=\frac{31}{20}$

Answer:

$x = \frac{3}{2}$ or $x = \frac{9}{2}$