Respuesta :
Use the properties of the real numbers to simplify each side of each equation. Then, use the properties of equalities to solve for x.
1)
[tex]-48(x-24)+12(x+4)=x+15[/tex]Use the distributive property to expand each parenthesis on the left hand side of the equation:
[tex]\begin{gathered} \Rightarrow-48(x)-48(-24)+12(x+4)=x+15 \\ \Rightarrow-48(x)-48(-24)+12(x)+12(4)=x+15 \end{gathered}[/tex]Simplify each term by multiplying the coefficients times the expressions inside the parenthesis:
[tex]\begin{gathered} \Rightarrow-48x-48(-24)+12(x)+12(4)=x+15 \\ \Rightarrow-48x+1152+12(x)+12(4)=x+15 \\ \Rightarrow-48x+1152+12x+12(4)=x+15 \\ \Rightarrow-48x+1152+12x+48=x+15 \end{gathered}[/tex]Use the commutative property to change the order of the terms on the left hand side of the equation to bring like terms together. Then, add them:
[tex]\begin{gathered} \Rightarrow-48x+12x+1152+48=x+15 \\ \Rightarrow-36x+1152+48=x+15 \\ \Rightarrow-36x+1200=x+15 \end{gathered}[/tex]Substract x from each side of the equation:
[tex]\begin{gathered} \Rightarrow-36x+1200-x=x+15-x \\ \Rightarrow-37x+1200=15 \end{gathered}[/tex]Substract 1200 from each side of the equation:
[tex]\begin{gathered} \Rightarrow-37x+1200-1200=15-1200 \\ \Rightarrow-37x=-1185 \end{gathered}[/tex]Divide both sides by -37:
[tex]\begin{gathered} \Rightarrow-\frac{37x}{-37}=-\frac{1185}{-37} \\ \Rightarrow x=\frac{1185}{37} \end{gathered}[/tex]Since 1185/37 is a rational number, then this equation has a rational solution.
2)
[tex]12(x+2)-4x=4(2x+11)-20[/tex]
Use the distributive property to expand the parenthesis on each side of the equation, and simplify the resulting expressions:
[tex]\begin{gathered} \Rightarrow12(x)+12(2)-4x=4(2x)+4(11)-20 \\ \Rightarrow12x+24-4x=8x+44-20 \\ \Rightarrow8x+24=8x+24 \end{gathered}[/tex]Since the expression 8x+24=8x+24 is an identity (the coefficients and constant terms are the same on both sides), then any number is a solution for this equation on the variable x.
Therefore, this equations has infinitely many solutions.
