{4x+3<2x+5 {6x-7≤7x+2

To solve the two inequalities given, let’s break them down one at a time.

Inequality 1:

\[ 4x + 3 < 2x + 5 \]

Step 1: Subtract \(2x\) from both sides: \[ 4x - 2x + 3 < 5 \] \[ 2x + 3 < 5 \]

Step 2: Subtract \(3\) from both sides: \[ 2x < 5 - 3 \] \[ 2x < 2 \]

Step 3: Divide by \(2\): \[ x < 1 \]

Inequality 2:

\[ 6x - 7 \leq 7x + 2 \]

Step 1: Subtract \(6x\) from both sides: \[ -7 \leq 7x - 6x + 2 \] \[ -7 \leq x + 2 \]

Step 2: Subtract \(2\) from both sides: \[ -7 - 2 \leq x \] \[ -9 \leq x \] or equivalently \( x \geq -9 \)

Solution Summary:

1. From the first inequality, we have: \( x < 1 \)
2. From the second inequality, we have: \( x \geq -9 \)

Combined Solution:

The solution to the compound inequalities is: \[ -9 \leq x < 1 \]

This means that \(x\) can take any value greater than or equal to \(-9\) but less than \(1\).