Problem 88 - Algebra 1: Concepts and Skills

Question

Solve the inequality. \(7+3 x \geq-2\)

Step-by-Step Solution

Answer

The solution for the given inequality is \(x \geq -3\). All values greater than or equal to -3 are solutions for the inequality.

Step 1: Isolate the Variable

Subtract 7 from both sides of the inequality to isolate \(3x\) on one side. This gives \(3x \geq -2 - 7\) or \(3x \geq -9\).

Step 2: Divide by Coefficient of x

Now, divide both sides of the inequality by 3 (the coefficient of x) to solve for x. This results in \(x \geq -9 / 3\) or \(x \geq -3\).

Step 3: Write the Final Answer

The final answer in decimal form is \(x \geq -3\). This means that all values greater than or equal to -3 are solutions for the inequality.

Key Concepts

Isolating the Variable

To solve an inequality like \(7+3x \geq -2\), we often start by isolating the variable, which is \(x\) in this case. This means we want all terms containing \(x\) on one side of the inequality and the constants on the other side.

Isolating the variable involves moving certain terms from one side of the inequality to the other. We can do this by adding or subtracting those terms. In our problem, we subtract 7 from both sides to get rid of the number 7 attached to the \(3x\). This gives us:

• \(3x \geq -9\)

The goal is to simplify the expression such that \(x\) stands alone on one side of the inequality. This makes it easier to identify the range or set of values \(x\) can take. Always remember, what you do to one side of the inequality, you must do to the other, to keep the inequality balanced.

Dividing by a Coefficient

After isolating the term with the variable, the next step is to address the coefficient attached to \(x\). In our example, the equation simplifies to \(3x \geq -9\). Here, 3 is the coefficient of \(x\).

To fully solve for \(x\), we need to remove this coefficient. We do this by dividing each side of the inequality by 3, the coefficient itself.

This yields a simpler form:

• \(x \geq \frac{-9}{3}\)
• Thus, \(x \geq -3\)

It's essential to remember that while dividing by a positive number like 3, the direction of the inequality sign remains the same. However, if you were dividing by a negative number, you would reverse the inequality sign.
This is a crucial rule when solving inequalities.

Inequality Solutions

Once we have isolated the variable and simplified the equation by dividing by its coefficient, we arrive at the solution of the inequality. In our case, it tells us that:

• \(x \geq -3\)

This means:\

• Any number greater than or equal to -3 satisfies the original inequality \(7 + 3x \geq -2\).

The solution to an inequality is a range of values that fit the conditions set by the inequality. In this case, \(x\) can take on any value from -3 to infinity, inclusive of -3.
Understanding inequality solutions helps in grasping the scope of possible values for \(x\). It is important in various mathematical contexts, as well as in real-world scenarios where such ranges are pivotal.