Solve x+1/2+x+2/3+x+3/4geq1+x/2

$$\frac{ x+1 }{ 2 } + \frac{ x+2 }{ 3 } + \frac{ x+3 }{ 4 } \geq 1+ \frac{ x }{ 2 }$$

Solve for x

$$x\geq -\frac{11}{7}$$

Solution Steps

Multiply both sides of the equation by 12, the least common multiple of 2,3,4. Since 12 is positive, the inequality direction remains the same.

$$6\left(x+1\right)+4\left(x+2\right)+3\left(x+3\right)\geq 12+6x$$

Use the distributive property to multiply 6 by x+1.

$$6x+6+4\left(x+2\right)+3\left(x+3\right)\geq 12+6x$$

Use the distributive property to multiply 4 by x+2.

$$6x+6+4x+8+3\left(x+3\right)\geq 12+6x$$

Combine 6x and 4x to get 10x.

$$10x+6+8+3\left(x+3\right)\geq 12+6x$$

Add 6 and 8 to get 14.

$$10x+14+3\left(x+3\right)\geq 12+6x$$

Use the distributive property to multiply 3 by x+3.

$$10x+14+3x+9\geq 12+6x$$

Combine 10x and 3x to get 13x.

$$13x+14+9\geq 12+6x$$

Add 14 and 9 to get 23.

$$13x+23\geq 12+6x$$

Subtract 6x from both sides.

$$13x+23-6x\geq 12$$

Combine 13x and -6x to get 7x.

$$7x+23\geq 12$$

Subtract 23 from both sides.

$$7x\geq 12-23$$

Subtract 23 from 12 to get -11.

$$7x\geq -11$$

Divide both sides by 7. Since 7 is positive, the inequality direction remains the same.

$$x\geq -\frac{11}{7}$$