What is the solution set of 2x−3=5 and 5x+3>−12 intersection of solutions.

Answer 1

To determine the solution set for the equations \( 2x - 3 = 5 \) and the inequality \( 5x + 3 > -12 \), we will solve each one separately and then find the intersection of their solutions.

Step 1: Solve the equation \( 2x - 3 = 5 \)

1. Add 3 to both sides: \[ 2x - 3 + 3 = 5 + 3 \] \[ 2x = 8 \]

2. Divide by 2: \[ x = \frac{8}{2} = 4 \]

So, the solution to the equation \( 2x - 3 = 5 \) is \( x = 4 \).

Step 2: Solve the inequality \( 5x + 3 > -12 \)

1. Subtract 3 from both sides: \[ 5x + 3 - 3 > -12 - 3 \] \[ 5x > -15 \]

2. Divide by 5: \[ x > -3 \]

So, the solution to the inequality \( 5x + 3 > -12 \) is \( x > -3 \).

Step 3: Find the intersection of the solutions

We have:

• From the equation: \( x = 4 \)
• From the inequality: \( x > -3 \)

Now we check if the solution from the equation \( x = 4 \) satisfies the inequality \( x > -3 \):

• Since \( 4 > -3 \) holds true, \( x = 4 \) is indeed within the range of \( x > -3 \).

Conclusion

The intersection of the solutions is: \[ { 4 } \]

Thus, the solution set of the given equation and inequality is \( { 4 } \).