Answer :
Certainly! Let's solve the system of inequalities step-by-step:
### Inequality 1: [tex]\(2x > 6x\)[/tex]
1. Subtract [tex]\(6x\)[/tex] from both sides:
[tex]\[ 2x - 6x > 0 \][/tex]
2. Simplify the left side:
[tex]\[ -4x > 0 \][/tex]
3. To isolate [tex]\(x\)[/tex], divide both sides by [tex]\(-4\)[/tex]. Remember that dividing by a negative number reverses the inequality sign:
[tex]\[ x < 0 \][/tex]
So, the solution to the first inequality is:
[tex]\[ x < 0 \][/tex]
### Inequality 2: [tex]\(3(x + 2) \geq -9\)[/tex]
1. Distribute [tex]\(3\)[/tex] on the left side:
[tex]\[ 3x + 6 \geq -9 \][/tex]
2. Subtract [tex]\(6\)[/tex] from both sides:
[tex]\[ 3x \geq -15 \][/tex]
3. To isolate [tex]\(x\)[/tex], divide both sides by [tex]\(3\)[/tex]:
[tex]\[ x \geq -5 \][/tex]
So, the solution to the second inequality is:
[tex]\[ x \geq -5 \][/tex]
### Intersection of Solutions
The solution to the system of inequalities is the intersection of the two individual solutions:
[tex]\[ x < 0 \quad \text{AND} \quad x \geq -5 \][/tex]
This means that [tex]\(x\)[/tex] must be both less than [tex]\(0\)[/tex] and greater than or equal to [tex]\(-5\)[/tex].
Therefore, the final solution is:
[tex]\[ -5 \leq x < 0 \][/tex]
In interval notation, the solution can be written as:
[tex]\[ [-5, 0) \][/tex]
So, the range of [tex]\(x\)[/tex] that satisfies both inequalities is from [tex]\(-5\)[/tex] (inclusive) to [tex]\(0\)[/tex] (exclusive).
### Inequality 1: [tex]\(2x > 6x\)[/tex]
1. Subtract [tex]\(6x\)[/tex] from both sides:
[tex]\[ 2x - 6x > 0 \][/tex]
2. Simplify the left side:
[tex]\[ -4x > 0 \][/tex]
3. To isolate [tex]\(x\)[/tex], divide both sides by [tex]\(-4\)[/tex]. Remember that dividing by a negative number reverses the inequality sign:
[tex]\[ x < 0 \][/tex]
So, the solution to the first inequality is:
[tex]\[ x < 0 \][/tex]
### Inequality 2: [tex]\(3(x + 2) \geq -9\)[/tex]
1. Distribute [tex]\(3\)[/tex] on the left side:
[tex]\[ 3x + 6 \geq -9 \][/tex]
2. Subtract [tex]\(6\)[/tex] from both sides:
[tex]\[ 3x \geq -15 \][/tex]
3. To isolate [tex]\(x\)[/tex], divide both sides by [tex]\(3\)[/tex]:
[tex]\[ x \geq -5 \][/tex]
So, the solution to the second inequality is:
[tex]\[ x \geq -5 \][/tex]
### Intersection of Solutions
The solution to the system of inequalities is the intersection of the two individual solutions:
[tex]\[ x < 0 \quad \text{AND} \quad x \geq -5 \][/tex]
This means that [tex]\(x\)[/tex] must be both less than [tex]\(0\)[/tex] and greater than or equal to [tex]\(-5\)[/tex].
Therefore, the final solution is:
[tex]\[ -5 \leq x < 0 \][/tex]
In interval notation, the solution can be written as:
[tex]\[ [-5, 0) \][/tex]
So, the range of [tex]\(x\)[/tex] that satisfies both inequalities is from [tex]\(-5\)[/tex] (inclusive) to [tex]\(0\)[/tex] (exclusive).
