Respuesta :
Answer:
1. no solutions
2. many (infinite) solutions
3. one solution: x = 0
4. one solution: x = -17
Step-by-step explanation:
Question 1
[tex]\begin{aligned}&\textsf{Given equation}: & 3(x+4)&=3x+11\\&\textsf{Distribute}: & 3x+12&=3x+11\\&\textsf{Subtract }3x \textsf{ from both sides}: & 12 &=11\end{aligned}[/tex]
As 12 ≠ 11 there are no solutions.
Question 2
[tex]\begin{aligned}&\textsf{Given equation}: & -2(x+3)&=-2x-6\\&\textsf{Distribute}: & -2x-6 &=-2x-6\\&\textsf{Add }2x \textsf{ to both sides}: & -6 &=-6\\&\textsf{Add }6 \textsf{ to both sides}: & 0 & = 0\\\end{aligned}[/tex]
As 0 = 0 there are infinite (many) solutions.
Question 3
[tex]\begin{aligned}&\textsf{Given equation}: & 4(x-1) & =\dfrac{1}{2}(x-8)\\&\textsf{Distribute}: & 4x-4 &=\dfrac{1}{2}x-4\\&\textsf{Add }4 \textsf{ to both sides}: & 4x & = \dfrac{1}{2}x\\&\textsf{Subtract } \dfrac{1}{2}x \textsf{ from both sides}: & \dfrac{3}{2}x & = 0\\&\textsf{Divide both sides by } \dfrac{3}{2}: & x & = 0\end{aligned}[/tex]
Therefore, there is one solution, x = 0.
Question 4
[tex]\begin{aligned}&\textsf{Given equation}: & 3x-7 & = 4+6+4x\\&\textsf{Simplify}: & 3x-7 & = 10+4x\\&\textsf{Swap sides}: & 10+4x & = 3x-7\\&\textsf{Subtract 10 from both sides}: & 4x & = 3x-17\\&\textsf{Subtract }3x \textsf{ from both sides}: & x & = -17\end{aligned}[/tex]
Therefore, there is one solution, x = -17.
Answer:
1. 3(x+4)=3x+11: No solutions.
2. -2(x+3)=-2x-6: Infinite solutions, any number can make the statement true.
3. 4(x-1) = 1/2(x-8): x= 0.
4. 3x-7=4+6 +4x: x= -17.
Step-by-step explanation:
Equation 1.
1. Write the expression.
[tex]3(x+4)=3x+11[/tex]
2. Simplify the left side of the equation by applying the associative property of multiplication.
This is the rule we're applying: [tex]a(b+c)=(a*b)+(a*c)[/tex].
[tex](3)(x)+(3)(4)=3x+11\\ \\3x+12=3x+11[/tex]
3. Substract 3x from both sides.
[tex]3x+12-3x=3x+11-3x\\ \\12=11[/tex]
4. Conclude.
As you may notice, we just obtained a false statement in subtitle 3, because 12 doesn't equal 11. Hence, this equation doesn't have solutions.
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Equation 2.
1. Write the expression.
[tex]-2(x+3)=-2x-6[/tex]
2. Simplify the left side of the equation by applying the associative property of multiplication.
[tex](-2)(x)+(-2)(3)=-2x-6\\ \\-2x-6=-2x-6[/tex]
3. Conclude.
Whenever we get an equation that has the same arguments on both sides, this means that there are infinite solutions to this equation, any value of x will make the equation true. Hence, the equation has infinite solutions.
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Equation 3.
1. Write the expression.
[tex]4(x-1) = 1/2(x-8)[/tex]
2. Simplify the both sides of the equation by applying the associative property of multiplication.
[tex](4)(x)+(4)(-1) = (\frac{1}{2} )(x)+(\frac{1}{2} )(-8)\\ \\4x-4=\frac{1}{2} x-4[/tex]
3. Add 4 to both sides.
[tex]4x-4+4=\frac{1}{2} x-4+4\\ \\4x=\frac{1}{2} x[/tex]
4. Substract [tex]\frac{1}{2} x[/tex] from both sides.
[tex]4x-\frac{1}{2} x=\frac{1}{2} x-\frac{1}{2} x\\ \\4x-\frac{1}{2} x=0\\ \\4x-0.5x=0\\ \\3.5x=0\\ \\x=\frac{0}{3.5 } \\ \\x=0[/tex]
5. Concluide.
The solution for this equation is x= 0.
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Equation 4.
1. Write the expression.
[tex]3x-7=4+6 +4x[/tex]
2. Simplify by completing the addition.
[tex]3x-7=10 +4x[/tex]
3. Add 7 to both sides.
[tex]3x-7+7=10 +4x+7\\ \\3x=17+4x[/tex]
4. Substract 4x from both sides.
[tex]3x-4x=17+4x-4x\\ \\-x=17[/tex]
5. Multiply bith sides by -1.
[tex](-1)(-x)=(17)(-1)\\ \\x=-17[/tex]
6. Conclude.
The solution of this equation is x= -17.
