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irina1246 [14]
3 years ago
14

Solve the equation. Check for extraneous solutions. Type your answers in the blanks. Show your work. 20 Points!!

Mathematics
1 answer:
alexira [117]3 years ago
8 0

|4x + 3| = 9 + 2x

Since the variable is on both sides of the equation, you would, at the end, check for extraneous solutions.

Extraneous solutions are solutions that do not work with the equation, therefore they are "extra" solutions and un-included in your final answer.

Start the problem by splitting the equation into two equations, a positive case and a negative case. Your two equations would look like:

  1. 4x + 3 = 9 + 2x {positive case}
  2. 4x + 3 = -(9 + 2x) {negative case}
<h2><u>---Solving the equations---</u></h2><h3>[POSITIVE CASE]</h3>

Let's solve for the positive case first. Start by subtracting 3 from both sides of the equation.

  • 4x + 3 = 9 + 2x becomes 4x = 6 + 2x

Now subtract 2x from both sides of the equation.

  • 4x = 6 + 2x becomes 2x = 6

Finish off the problem by dividing both sides by 2 to isolate the variable x.

  • 2x = 6 becomes x = 3.
<h2>---</h2><h3>[NEGATIVE CASE]</h3>

Now let's solve for x in the negative case. Start by distributing the negative sign (-) inside the parentheses.

  • 4x + 3 = -(9 + 2x) becomes 4x + 3 = -9 - 2x

Subtract 3 from both sides just like the positive case.

  • 4x + 3 = -9 - 2x becomes 4x = -12 - 2x

Now add 2x to both sides of the equation.

  • 4x = -12 - 2x becomes 6x = -12

Finish off the problem by dividing both sides by 6 to isolate the variable x.

  • 6x = -12 becomes x = -2.
<h2><u>---Checking for extraneous solutions---</u></h2><h3>[CHECKING X = 3]</h3>

To check for extraneous solutions, or solutions that do not work, substitute what you got for x back into the original absolute value equation: |4x + 3| = 9 + 2x. Substitute 3 and -2 into the equation. Let's start by substituting 3 for x.

  • |4x + 3| = 9 + 2x becomes |4(3) + 3| = 9 + 2(3)

Start by multiplying 4 and 3 together inside the absolute value symbols.

  • |4(3) + 3| = 9 + 2(3) becomes |(12) + 3| = 9 + 2(3)

Now multiply 2 and 3 together.

  • |(12) + 3| = 9 + 2(3) becomes |(12) + 3| = 9 + (6)

Add 12 and 3 together inside the absolute value symbols; also add 9 and 6 together.

  • |(12) + 3| = 9 + (6)  becomes |(15)| = (15), which is the same as 15 = 15.

15 = 15 is a true statement so this means that 3 is a solution to the absolute value equation, so it is not an extraneous solution.

<h2>---</h2><h3>[CHECKING X = -2]</h3>

Let's see if -2 is a solution or not - substitute -2 for x into the equation: |4x + 3| = 9 + 2x.

  • |4x + 3| = 9 + 2x becomes |4(-2) + 3| = 9 + 2(-2)

Multiply 4 and -2 inside the absolute value symbols.

  • |4(-2) + 3| = 9 + 2(-2) becomes |(-8) + 3| = 9 + 2(-2)

Multiply 2 and -2.

  • |(-8) + 3| = 9 + 2(-2) becomes |(-8) + 3| = 9 + (-4)

Add -8 and 3 inside the absolute value symbols; also add 9 and -4.

  • |(-8) + 3| = 9 + (-4) becomes |(-5)| = (5), which is the same as 5 = 5.

5 = 5 is a true statement so that means it is not an extraneous solution. After checking for extraneous solutions, we have come to the conclusion that the two answers for the equation --> I4x + 3I = 9 + 2x <-- are <u>x = 3 or x = 2</u>.

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Anna71 [15]

(8a^{-3})^{\frac{-2}{3}} = \frac{a^2}{4}

<em><u>Solution:</u></em>

<em><u>Given that,</u></em>

(8a^{-3})^{\frac{-2}{3}

We have to write in simplest form

<em><u>Use the following law of exponent</u></em>

(a^m)^n = a^{mn}

Using this, simplify the given expression

(8a^{-3})^{\frac{-2}{3}} = 8^{\frac{-2}{3}} \times a^{ -3 \times \frac{-2}{3}}\\\\Simplifying\ we\ get\\\\(8a^{-3})^{\frac{-2}{3}} = 8^{\frac{-2}{3}} \times a^2\\\\We\ know\ that\ 8 = 2^3\\\\Therefore\\\\(8a^{-3})^{\frac{-2}{3}} =2^3^{\frac{-2}{3}} \times a^2\\\\(8a^{-3})^{\frac{-2}{3}} =2^{-2} \times a^2\\\\(8a^{-3})^{\frac{-2}{3}} = \frac{a^2}{4}

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3 years ago
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a hexagon inscribed in a circle has three consecutive sides each of length 3 and three consecutive sides each of length 5. the c
allochka39001 [22]

According to given conditions, m+n is equal to 409.

Consider the diagram below.

In the hexagon ABCDEF, let

AB = BC = CD = 3;

DE = EF = FA = 5;

Arc BAF is equal to one-third of the circle's circumference.

Hence, ∠BCF = ∠BEF = 60°;

Similarly, ∠CBE = ∠CFE = 60°;

Let the point of intersection of BE and CF be P, BE and AD be Q and CF and AD be R.

∴ Δ EFP and Δ BCP are equilateral, and so Δ PQR is also equilateral.

Also, ∠ BAD and ∠ BED subtend the same arc and so do ∠ ABE and ∠ ADE.

∴ Δ ABQ is similar to Δ EDQ

\frac{AQ}{EQ}  = \frac{BQ}{DQ} = \frac{AB}{ED} = \frac{3}{5}

Also,

\frac{\frac{AD - PQ}{2} }{PQ + 5} = \frac{3}{5}  

and \frac{3 - PQ}{\frac{AD + PQ}{2} }  = \frac{3}{5}

On solving these simultaneous equations, we get AD = 360/49

∴ m + n = 409.

To learn more about similarity of triangles, refer to this link:

brainly.com/question/25882965

#SPJ4

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1 year ago
Andre wants to save $40 to buy a gift for his dad. Andre’s neighbor will pay him weekly to mow the lawn, but Andre always gives
Dmitry_Shevchenko [17]

(I could answer more confidently if I had more context to the question)

I believe the answer to the question is Andre's neighbor pays Andre 10 dollars a week to mow his lawn.

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3 years ago
What is 10+2p=12+1.5p <br> p=
Allushta [10]

Answer:

p=4

Step-by-step explanation:

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