QuestionSolve the equations: 1) $\frac{5 x-3}{4}=\frac{2-4 x}{3}$ and 2) $\frac{1}{3}(x-2)-\frac{x}{2}>\frac{1}{6}$ .

Studdy Solution

STEP 1

Assumptions1. The variables in the equations are real numbers. . The equations are linear, meaning the variables are not raised to any power other than1.
3. The equations are equalities and inequalities, meaning we are looking for values of the variables that make the equations true.

STEP 2

For the first equation, $\frac{5 x-}{4}=\frac{2-4 x}{}$ , we will first clear the fractions by multiplying each side of the equation by the least common multiple (LCM) of the denominators (4 and).
$4 \times \times \left(\frac{5 x-}{4}\right) =4 \times \times \left(\frac{2-4 x}{}\right)$

STEP 3

implify the equation.
$3(5x -3) =(2 -x)$

STEP 4

Expand both sides of the equation.
$15x -9 =8 -16x$

STEP 5

Rearrange the equation to get all terms involving x on one side and the constant terms on the other side.
$15x +16x =8 +9$

STEP 6

implify the equation.
$31x =17$

STEP 7

olve for x by dividing both sides of the equation by31.
$x = \frac{17}{31}$

STEP 8

For the second equation, $\frac{1}{3}(x-2)-\frac{x}{2}>\frac{1}{6}$ , we will first clear the fractions by multiplying each side of the inequality by the least common multiple (LCM) of the denominators (3,2, and6).
$6 \times \left(\frac{1}{3}(x-2)\right) -6 \times \left(\frac{x}{2}\right) >6 \times \left(\frac{1}{6}\right)$

STEP 9

implify the inequality.
$2(x -2) -3x >$

STEP 10

Expand and simplify the inequality.
$2x -4 -3x >$

STEP 11

Rearrange the inequality to get all terms involving x on one side and the constant terms on the other side.
$-x > +4$

STEP 12

implify the inequality.
$-x >5$

STEP 13

Multiply both sides of the inequality by - to solve for x. Remember that when you multiply or divide both sides of an inequality by a negative number, the direction of the inequality sign changes.
$x < -5$ So, the solutions are $x = \frac{17}{31}$ and $x < -5$ .

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QuestionSolve the equations: 1) $\frac{5 x-3}{4}=\frac{2-4 x}{3}$ and 2) $\frac{1}{3}(x-2)-\frac{x}{2}>\frac{1}{6}$ .

Studdy Solution

STEP 1

Assumptions1. The variables in the equations are real numbers. . The equations are linear, meaning the variables are not raised to any power other than1.
3. The equations are equalities and inequalities, meaning we are looking for values of the variables that make the equations true.

STEP 2

For the first equation, $\frac{5 x-}{4}=\frac{2-4 x}{}$ , we will first clear the fractions by multiplying each side of the equation by the least common multiple (LCM) of the denominators (4 and).
$4 \times \times \left(\frac{5 x-}{4}\right) =4 \times \times \left(\frac{2-4 x}{}\right)$

STEP 3

implify the equation.
$3(5x -3) =(2 -x)$

STEP 4

Expand both sides of the equation.
$15x -9 =8 -16x$

STEP 5

Rearrange the equation to get all terms involving x on one side and the constant terms on the other side.
$15x +16x =8 +9$

STEP 6

implify the equation.
$31x =17$

STEP 7

olve for x by dividing both sides of the equation by31.
$x = \frac{17}{31}$

STEP 8

STEP 9

implify the inequality.
$2(x -2) -3x >$

STEP 10

Expand and simplify the inequality.
$2x -4 -3x >$

STEP 11

Rearrange the inequality to get all terms involving x on one side and the constant terms on the other side.
$-x > +4$

STEP 12

implify the inequality.
$-x >5$

STEP 13

Multiply both sides of the inequality by - to solve for x. Remember that when you multiply or divide both sides of an inequality by a negative number, the direction of the inequality sign changes.
$x < -5$ So, the solutions are $x = \frac{17}{31}$ and $x < -5$ .

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QuestionSolve the equations: 1) 5x−34=2−4x3\frac{5 x-3}{4}=\frac{2-4 x}{3}45x−3​=32−4x​ and 2) 13(x−2)−x2>16\frac{1}{3}(x-2)-\frac{x}{2}>\frac{1}{6}31​(x−2)−2x​>61​.

Studdy Solution

STEP 1

Assumptions1. The variables in the equations are real numbers. . The equations are linear, meaning the variables are not raised to any power other than1.3. The equations are equalities and inequalities, meaning we are looking for values of the variables that make the equations true.

STEP 2

STEP 3

implify the equation.3(5x−3)=(2−x)3(5x -3) =(2 -x)3(5x−3)=(2−x)

STEP 4

Expand both sides of the equation.15x−9=8−16x15x -9 =8 -16x15x−9=8−16x

STEP 5

Rearrange the equation to get all terms involving x on one side and the constant terms on the other side.15x+16x=8+915x +16x =8 +915x+16x=8+9

STEP 6

implify the equation.31x=1731x =1731x=17

STEP 7

olve for x by dividing both sides of the equation by31.x=1731x = \frac{17}{31}x=3117​

STEP 8

STEP 9

implify the inequality.2(x−2)−3x>2(x -2) -3x >2(x−2)−3x>

STEP 10

Expand and simplify the inequality.2x−4−3x>2x -4 -3x >2x−4−3x>

STEP 11

Rearrange the inequality to get all terms involving x on one side and the constant terms on the other side.−x>+4-x > +4−x>+4

STEP 12

implify the inequality.−x>5-x >5−x>5

STEP 13

Multiply both sides of the inequality by - to solve for x. Remember that when you multiply or divide both sides of an inequality by a negative number, the direction of the inequality sign changes.x<−5x < -5x<−5So, the solutions are x=1731x = \frac{17}{31}x=3117​ and x<−5x < -5x<−5.

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QuestionSolve the equations: 1) $\frac{5 x-3}{4}=\frac{2-4 x}{3}$ and 2) $\frac{1}{3}(x-2)-\frac{x}{2}>\frac{1}{6}$ .

Assumptions1. The variables in the equations are real numbers. . The equations are linear, meaning the variables are not raised to any power other than1.
3. The equations are equalities and inequalities, meaning we are looking for values of the variables that make the equations true.

implify the equation.
$3(5x -3) =(2 -x)$

Expand both sides of the equation.
$15x -9 =8 -16x$

Rearrange the equation to get all terms involving x on one side and the constant terms on the other side.
$15x +16x =8 +9$

implify the equation.
$31x =17$

olve for x by dividing both sides of the equation by31.
$x = \frac{17}{31}$

implify the inequality.
$2(x -2) -3x >$

Expand and simplify the inequality.
$2x -4 -3x >$

Rearrange the inequality to get all terms involving x on one side and the constant terms on the other side.
$-x > +4$

implify the inequality.
$-x >5$

Multiply both sides of the inequality by - to solve for x. Remember that when you multiply or divide both sides of an inequality by a negative number, the direction of the inequality sign changes.
$x < -5$ So, the solutions are $x = \frac{17}{31}$ and $x < -5$ .