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mina [271]
3 years ago
7

et f(x) = x + |2x − 3|. Find all solutions to the equation f(x) = 12. (Enter your answers as a comma-separated list. If an answe

r does not exist, enter DNE.) x = $$5,−9 (b) Let g(x) = 5x − 3 + |x + 4|. Find all values of a which satisfy the equation g(a) = 4a + 9. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) a = $$4 (c) Let h(x) = |x| − 2x + 5. Find all solutions to the equation h(x − 1) = x − 3. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) x = $$ 9 2​
Mathematics
1 answer:
Alinara [238K]3 years ago
8 0

Answer:

1. {5,-9}

2.{4}

3.{9/2}

Step-by-step explanation:

The equation f(x)=x+\lvert 2x-3 \rvert=12 can be solved as follows:

x+\lvert 2x-3 \rvert =12\,\,\,\text{(given equation)}

\lvert 2x-3 \rvert =12-x

2x-3=12-x\,\,\text{or}\,\,2x-3=-(12-x)

3x=15\,\,\text{or}\,\,\x=-9

Then, the solutions of the equation are x=5, x=-9.

Now, if g(x)=5x-3+\lvert x+4\rvert, we can solve the equation g(a)=4a+9 as follows:

5a-3+\lvert a+4 \rvert =4a+9

\lvert a+4\rvert =4a+9-5a+3

a+4 =-a+12\,\,\text{or}\,\,a+4=-(-a+12)

2a=8\,\,\text{or}\,\,4=12

So, the unique solution of this last equation is a=4

The last equation is h(x-1)=x-3, where h(x)=\lvert x \rvert -2x+5.. To solve this equation we can proceed as follows:

h(x-1)=x-3

\lvert x-1\rvert -2(x-1)+5=x-3

\lvert x-1 \rvert -2x+2+5=x-3

\lvert x-1\rvert =3x-10

x-1=3x-10\,\,\text{or}\,\,x-1=-(3x-10)

-2x=-9\,\,\text{or}\,\,4x=11

x=9/2\,\,\text{or}\,\,x=11/4

But, x=11/4 is not a solution of this equation. So the unique solution is x=9/2.

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