Solve the following inequality for [tex]\( x \)[/tex]:

[tex]\[ \begin{array}{l}
2x \ \textgreater \ 6x \\
3(x + 2) \geq -9
\end{array} \][/tex]



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).