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How to Solve the Two-Step Linear Inequality $-2 \\geq -7 + \\frac{x}{6}$ for x
Mathematics
Grade 8 (Junior High School)
Question Content
Solve the inequality for x: $-2 \\geq -7 + \\frac{x}{6}$. Simplify your answer as much as possible.
Correct Answer
$x \\leq 30$
Detailed Solution Steps
1
Step 1: Isolate the term with x by adding 7 to both sides of the inequality to cancel out the -7 on the right side: $-2 + 7 \\geq -7 + 7 + \\frac{x}{6}$, which simplifies to $5 \\geq \\frac{x}{6}$.
2
Step 2: Eliminate the fraction by multiplying both sides of the inequality by 6: $5 \\times 6 \\geq \\frac{x}{6} \\times 6$, which simplifies to $30 \\geq x$.
3
Step 3: Rewrite the inequality in standard variable-first form: $x \\leq 30$.
Knowledge Points Involved
1
Two-step linear inequalities
Linear inequalities that require two inverse operations (e.g., addition/subtraction and multiplication/division) to isolate the variable. They follow the same logic as two-step equations, with the key rule that multiplying/dividing by a negative number reverses the inequality sign (not needed here since we multiply by a positive 6).
2
Inverse operations for inequalities
Operations used to isolate the variable in an inequality: adding/subtracting the same number to both sides preserves the inequality direction; multiplying/dividing both sides by a positive number preserves the direction, while multiplying/dividing by a negative number reverses the inequality sign.
3
Inequality notation and rearrangement
Inequalities can be rewritten to place the variable on the left side for clarity (e.g., $30 \\geq x$ is equivalent to $x \\leq 30$) without changing the meaning, as long as the inequality symbol points correctly to the same value.
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