- Identify the distribution step by applying \(a(b+c) = ab + ac\) to the left side: \(-4(3-5x) = -12 + 20x\).
- Recognize that distribution does not change the inequality direction, so the symbol \(\geq\) is preserved.
- Form the resulting inequality \(-12 + 20x \geq -6x + 9\).
- Select the matching option: \(\boxed{-12+20x \geq -6x+9}\).
Explanation
- Analyzing the Inequality
The objective is to determine the correct first step in solving the linear inequality \(-4(3-5x) \geq -6x+9\). To solve this, we apply the distributive property on the left-hand side of the inequality. The distributive property states that for any real numbers \(a, b,\) and \(c\), the expression \(a(b+c)\) is equivalent to \(ab + ac\). Here, \(a = -4\), \(b = 3\), and \(c = -5x\).
- Distributing the Terms
We calculate the distribution as follows:
\[-4(3-5x) = (-4 \cdot 3) + (-4 \cdot -5x)\]
The first part of the expression is \(-4 \cdot 3 = -12\).
The second part of the expression involves multiplying two negative numbers: \(-4 \cdot -5x = 20x\).
Combining these results, the expression becomes \(-12 + 20x\).
Crucially, when distributing a value, we do not change the inequality sign unless we multiply or divide the entire inequality by a negative number. Since we are simply simplifying one side, the inequality symbol \(\geq\) remains unchanged.
- Conclusion and Identification of the Correct Step
Substituting the simplified expression back into the original inequality, we obtain:
\[-12 + 20x \geq -6x + 9\]
Comparing this result to the provided options:
- \(-12-20x \leq -6x+9\)
- \(-12-20x \geq -6x+9\)
- \(-12+20x \leq -6x+9\)
- \(-12+20x \geq -6x+9\)
The correct option is \(-12+20x \geq -6x+9\). Thus, the correct first step in solving the inequality is \(-12+20x \geq -6x+9\). Final result is \(\boxed{-12+20x \geq -6x+9}\).
Examples
Distributive property is frequently used in daily life when calculating bulk costs. For example, if you are buying 4 items priced at \$3 each and a variable amount of other items priced at -\$5 (representing a discount per unit), the total cost adjustment could be represented by \(-4(3-5x)\). Simplifying this helps quickly determine how the total price changes as the quantity \(x\) of the discounted items varies. Mastering this algebraic step ensures accuracy in financial modeling and resource management.