QuestionSolve the equation $3|x-4|=|2x+5|$ .

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 is $3|x-4|=|x+5|$ .

STEP 2

We can solve this equation by considering the two possible cases separately. The first case is when $x-4$ and $2x+5$ are both positive or zero, and the second case is when $x-4$ and $2x+5$ are both negative.

STEP 3

For the first case, we can write the equation as $3(x-) =2x+5$

STEP 4

implify the equation by distributing the3 on the left side and combining like terms $3x -12 =2x +$

STEP 5

Subtract $2x$ from both sides to get $x -12 =5$

STEP 6

Add12 to both sides to solve for $x$ $x =5 +12$

STEP 7

Calculate the value of $x$ $x =17$

STEP 8

Now, we need to check if $x=17$ is a valid solution for the original equation. Substitute $x=17$ into the original equation $3|17-4| = |2(17)+5|$

STEP 9

Calculate the absolute values $3|13| = |34+5|$

STEP 10

Calculate the values inside the absolute value signs $39 =39$

STEP 11

Since the equation is true, $x=17$ is a valid solution for the first case.

STEP 12

For the second case, we can write the equation as $(-x+4) = -(2x+5)$

STEP 13

implify the equation by distributing the3 on the left side and combining like terms $-3x +12 = -2x -5$

STEP 14

Add $3x$ to both sides to get $12 = x -$

STEP 15

Add5 to both sides to solve for $x$ $x =12 +5$

STEP 16

Calculate the value of $x$ $x =$

STEP 17

Now, we need to check if $x=17$ is a valid solution for the original equation. Substitute $x=17$ into the original equation $3|17-4| = |2(17)+5|$

STEP 18

Calculate the absolute values $3|13| = |34+5|$

STEP 19

Calculate the values inside the absolute value signs $39 =39$

STEP 20

Since the equation is true, $x=17$ is also a valid solution for the second case.
Therefore, the solution to the equation $3|x-4|=|x+5|$ is $x=17$ .

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