Question

practice
how many solutions?

1. lin was looking at the equation (2x - 32 + 4(3x - 2.462)=14x). she said,

\i can tell right away there are no solutions, because on the left side, you
will have (2x + 12x) and a bunch of constants, but you have just (14x) on the
right side.\ do you agree with lin? explain your reasoning

1. han was looking at the equation (6x - 4 + 2(5x + 2)=16x). he said, \i can

tell right away there are no solutions, because on the left side, you will have
(6x + 10x) and a bunch of constants, but you have just (16x) on the right side.\
do you agree with han? explain your reasoning.

1. decide whether each equation is true for all, one, or no values of (x).

a. (6x - 4 = -4 + 6x)
b. (4x - 6 = 4x + 3)
c. (-2x + 4 = 3x + 4)

1. solve each of these equations. explain or show your reasoning. (lesson 4 - 4)

a. (3|x - 5| = 6)
b. (2left(x - \frac{7}{3}
ight)=0)
c. (4x - 5 = 2 - x)

Explanation:

Step1: Expand left side of Lin's eq

$2x - 32 + 4(3x - 2.462) = 2x - 32 + 12x - 9.848$

Step2: Combine like terms

$2x + 12x - 32 - 9.848 = 14x - 41.848$

Step3: Compare to right side

$14x - 41.848 = 14x$ simplifies to $-41.848=0$, false.

Step4: Expand left side of Han's eq

$6x - 4 + 2(5x + 2) = 6x - 4 + 10x + 4$

Step5: Combine like terms

$6x + 10x - 4 + 4 = 16x$

Step6: Compare to right side

$16x = 16x$ is always true.

Step7: Analyze 3a: Rearrange terms

$6x - 4 = -4 + 6x$ simplifies to $6x-4=6x-4$, always true.

Step8: Analyze 3b: Cancel 4x terms

$4x - 6 = 4x + 3$ simplifies to $-6=3$, false.

Step9: Analyze 3c: Solve for x

$-2x + 4 = -3x + 4$ → $-2x + 3x = 4 - 4$ → $x=0$

Step10: Solve 4a: Isolate $(x-5)$

$3(x-5)=6$ → $x-5=\frac{6}{3}=2$

Step11: Solve for x in 4a

$x=2+5=7$

Step12: Solve 4b: Isolate $(x-\frac{2}{3})$

$2(x-\frac{2}{3})=0$ → $x-\frac{2}{3}=0$

Step13: Solve for x in 4b

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

Step14: Solve 4c: Combine like terms

$4x - 5 = 2 - x$ → $4x + x = 2 + 5$ → $5x=7$

Step15: Solve for x in 4c

$x=\frac{7}{5}$

Answer:

1. Agree with Lin. Simplifying the left side gives $14x - 41.848$, which can never equal $14x$ (since $-41.848

eq 0$), so there are no solutions.

1. Disagree with Han. Simplifying the left side gives $16x$, which equals the right side $16x$ for all values of $x$, so there are infinitely many solutions.

3.
a. True for all values of $x$
b. True for no values of $x$
c. True for one value of $x$ ($x=0$)
4.
a. $x=7$
b. $x=\frac{2}{3}$
c. $x=\frac{7}{5}$