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10-116.

Solve the following equations and inequalities, if possible. Represent each solution on a number line.

$\frac{3}{9}-\frac{x}{3}=\frac{x}{5}$

Step 1(a):

Multiply the equation by a common multiple of the denominators to remove the fractions.

$45\left(\frac{3}{9}-\frac{x}{3}=\frac{x}{5}\right)$

$15-15x=9x$

Step 2(a):

Solve for $x$ .

Answer (a):

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

$\left(3+x\right)^2<9$
Step 1(b):
Find the boundary point.
$\left(3+x\right)^2=9$
Step 2(b):
Take the square root of both sides.
$3+x=\pm3$
Step 3(b):
Solve for $x$ .
Step 4(b):
Test numbers on all sides of the boundary points.
$(3+(-7))^2<9(3+(-1))^2<9(3+(1))^2<9$
$(-4)^2<92^2<94^2<9$
$16<94<916<9$
False True False
Answer (b):

$8\left|x+1\right|\ge64$
Hint 1(c):
Divide both sides by $8$ .
Hint 2(c):
The absolute value of what two numbers equals $8$ ?
Hint 3(c):
Follow the steps in part (b).

$11-\sqrt{x+3}=13$
Step 1(d):
Subtract $11$ from both sides.
Step 2(d):
Square both sides.
Step 3(d):
Subtract $3$ from both sides.

$\frac { x } { 8 } = \frac { 2 } { x }$
Hint 1(e):
Follow the steps in part (a).
Hint 2(e):
A common multiple of $x$ and $8$ is $8x$ .

$\left|x-5\right|+1>0$
Hint (f):
Follow the steps in part (c).

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