Question

summary

• to solve for a variable in an algebraic equation, perform the same operation on both sides of the equation to isolate the variable on one side. simplify the other side of the equation by combining similar terms.
• use transposition, or moving a term to the other side of the equation and changing it to its mirror image, as a shortcut.
• to transpose a term that is added or subtracted, use the opposite sign (+ or -) when the term is moved to the other side of the equation.
• to transpose a term that is multiplied or divided, multiply by its reciprocal.
• to solve inequalities, follow the same procedure as for equations. if multiplying or dividing by a negative value, reverse the inequality sign.

exercise 1
directions: choose the best answer from the four choices given. answers are on page 281.

1. if $5(2x - 5)=10$, then $x =$?

a -2.5
b -1.5
c 2
d 3.5

1. if $\frac{-(-5x + 36)}{2}=2$, then $x =$?

a -8
b $-\frac{32}{5}$
c $\frac{32}{5}$
d 8

1. if $\frac{-2}{3}=\frac{12}{x}$, then $x =$?

a -24
b -18
c -16
d 18

1. if $2x + 3>9$, which of the following can be the value of $x$?

a -4
b -3
c 3
d 4

1. which of the following is equivalent to $-3x + 6>x - 10$?

a $x > -12$
b $x < -4$
c $x < 4$
d $x > 4$

Explanation:

Problem 1

Step1: Divide both sides by 5

$\frac{5(2x - 5)}{5}=\frac{10}{5}$
$2x - 5 = 2$

Step2: Add 5 to both sides

$2x - 5 + 5 = 2 + 5$
$2x = 7$

Step3: Divide by 2

$\frac{2x}{2}=\frac{7}{2}$
$x = 3.5$

Wait, earlier miscalculation. Wait, original equation: $5(2x - 5)=10$. Divide both sides by 5: $2x -5 = 2$. Then add 5: $2x = 7$? No, 2 +5 is 7? Wait, 10 divided by 5 is 2. Then 2x -5 =2. So 2x=2+5=7? Then x=3.5. But let's check again. Wait, maybe I made a mistake. Wait, let's re - solve:

Step1: Expand the left - hand side

$5\times2x-5\times5 = 10$
$10x-25 = 10$

Step2: Add 25 to both sides

$10x-25 + 25=10 + 25$
$10x=35$

Step3: Divide by 10

$x=\frac{35}{10}=3.5$

Step1: Simplify the numerator

$-(-5x + 36)=5x-36$ (since negative of a negative is positive, and negative of positive 36 is - 36)
So the equation becomes $\frac{5x - 36}{2}=2$

Step2: Multiply both sides by 2

$5x-36=2\times2$
$5x-36 = 4$

Step3: Add 36 to both sides

$5x-36 + 36=4 + 36$
$5x=40$

Step4: Divide by 5

$x=\frac{40}{5}=8$

Step1: Cross - multiply

$\frac{-2}{3}=\frac{12}{x}$ implies $-2\times x=3\times12$

Step2: Simplify the right - hand side

$-2x = 36$

Step3: Divide by - 2

$x=\frac{36}{-2}=-18$

Answer:

D. 3.5

Problem 2