How do you solve #1/2x+3<-2x-7 #?

To solve the inequality 1/2x + 3 < -2x - 7, first, we'll isolate the variable x by performing the necessary operations to both sides of the inequality.

1/2x + 3 < -2x - 7

Subtract 3 from both sides:

1/2x < -2x - 10

Multiply both sides by 2 to eliminate the fraction:

x < -4x - 20

Add 4x to both sides:

5x < -20

Divide both sides by 5:

x < -4

To solve the inequality ( \frac{1}{2}x + 3 < -2x - 7 ), follow these steps:

1. Begin by isolating the variable term on one side of the inequality.
2. Perform any necessary operations to simplify the inequality.
3. Solve for the variable.
4. Check if the solution is valid by ensuring that it satisfies the original inequality.

Now, let's solve the given inequality:

[ \frac{1}{2}x + 3 < -2x - 7 ]

Step 1: Isolate the variable term on one side of the inequality:

[ \frac{1}{2}x + 2x < -7 - 3 ]

Step 2: Combine like terms and simplify:

[ \frac{5}{2}x < -10 ]

Step 3: Divide both sides of the inequality by ( \frac{5}{2} ):

[ x < -10 \times \frac{2}{5} ]

[ x < -4 ]

Step 4: Check the solution by substituting a value of ( x ) into the original inequality. Any value less than -4 should satisfy the inequality.

Therefore, the solution to the inequality ( \frac{1}{2}x + 3 < -2x - 7 ) is ( x < -4 ).