Question

select all the expressions that are equivalent to $x + \frac{3x}{2} + \frac{x}{4}$.$\frac{x}{4}(4 + 6 + 1)LXB0x + \frac{2x}{3}LXB1\frac{11x}{4}$

Explanation:

Step1: Simplify the original expression

First, rewrite $x$ as $\frac{4x}{4}$ and $\frac{3x}{2}$ as $\frac{6x}{4}$ to get a common denominator:
$$x + \frac{3x}{2} + \frac{x}{4} = \frac{4x}{4} + \frac{6x}{4} + \frac{x}{4}$$
Combine the fractions:
$$=\frac{4x + 6x + x}{4} = \frac{11x}{4}$$

Step2: Evaluate the first option

Simplify $\frac{x}{4}(4+6+1)$:
Calculate the sum inside the parentheses: $4+6+1=11$
$$\frac{x}{4}(11) = \frac{11x}{4}$$

Step3: Evaluate the second option

Simplify $x(1+1.5+0.25)$:
Calculate the sum inside the parentheses: $1+1.5+0.25=2.75$
Convert 2.75 to a fraction: $2.75 = \frac{11}{4}$, so:
$$x \times \frac{11}{4} = \frac{11x}{4}$$

Step4: Evaluate the third option

Simplify $x + \frac{2x}{3}$:
Rewrite $x$ as $\frac{3x}{3}$:

$$\frac{3x}{3} + \frac{2x}{3} = \frac{5x}{3} eq \frac{11x}{4}$$

Step5: Evaluate the fourth option

$\frac{5x}{6}$ is already simplified, and $\frac{5x}{6}
eq \frac{11x}{4}$

Step6: Evaluate the fifth option

$\frac{11x}{4}$ matches the simplified original expression.

Answer:

• $\frac{x}{4}(4 + 6 + 1)$
• $x(1 + 1.5 + 0.25)$
• $\frac{11x}{4}$