Question

1. $\frac{3x}{2}-4=\frac{x}{3}+2$ 18. $0.5x + 6 = 0.25x+9$ 20. $4(x + 1)=3(x + 4)-2$ 22. $3(x + 2)=3x + 6$ 24. $4(x + 1)=4x + 4$ challenge: 26. the perimeter of a rectangle is 48 cm. write and solve an equation to find the value of

Explanation:

1. Solve equation 16: $\frac{3x}{2}-4=\frac{x}{3}+2$

• Step 1: Get rid of the fractions by multiplying through by the least - common multiple (LCM) of 2 and 3, which is 6.
• $6\times(\frac{3x}{2}-4)=6\times(\frac{x}{3}+2)$
• Using the distributive property $a(b + c)=ab+ac$, we have $6\times\frac{3x}{2}-6\times4 = 6\times\frac{x}{3}+6\times2$.
• This simplifies to $9x-24 = 2x + 12$.
• Step 2: Isolate the variable terms on one side and the constant terms on the other side.
• Subtract $2x$ from both sides: $9x-2x-24=2x-2x + 12$, which gives $7x-24 = 12$.
• Then add 24 to both sides: $7x-24 + 24=12 + 24$, resulting in $7x=36$.
• Step 3: Solve for $x$.
• Divide both sides by 7: $x=\frac{36}{7}$.

1. Solve equation 18: $0.5x + 6=0.25x + 9$

• Step 1: Subtract $0.25x$ from both sides.
• $0.5x-0.25x+6=0.25x-0.25x + 9$.
• $0.25x+6 = 9$.
• Step 2: Subtract 6 from both sides.
• $0.25x+6 - 6=9 - 6$.
• $0.25x=3$.
• Step 3: Since $0.25=\frac{1}{4}$, we can rewrite the equation as $\frac{1}{4}x = 3$. Multiply both sides by 4 to solve for $x$.
• $x = 12$.

1. Solve equation 20: $4(x + 1)=3(x + 4)-2$

• Step 1: Expand both sides using the distributive property.
• $4x+4=3x + 12-2$.
• $4x+4=3x + 10$.
• Step 2: Subtract $3x$ from both sides.
• $4x-3x+4=3x-3x + 10$.
• $x+4 = 10$.
• Step 3: Subtract 4 from both sides.
• $x=6$.

1. Solve equation 22: $3(x + 2)=3x + 6$

• Step 1: Expand the left - hand side using the distributive property.
• $3x+6=3x + 6$.
• Subtracting $3x$ from both sides gives $3x-3x+6=3x-3x + 6$, which simplifies to $6 = 6$. This is an identity, and the solution is all real numbers.

1. Solve equation 24: $4(x + 1)=4x + 4$

• Step 1: Expand the left - hand side using the distributive property.
• $4x+4=4x + 4$.
• Subtracting $4x$ from both sides gives $4x-4x+4=4x-4x + 4$, which simplifies to $4 = 4$. This is an identity, and the solution is all real numbers.

Answer:

1. $x=\frac{36}{7}$
2. $x = 12$
3. $x=6$
4. All real numbers
5. All real numbers