Solve for $x$: $4-(x+2)<-3(x+4)$
Answer (2)
Simplify the inequality by distributing and combining like terms.
Isolate the variable x on one side of the inequality.
Divide both sides by the coefficient of x to solve for x .
The solution to the inequality 4 − ( x + 2 ) < − 3 ( x + 4 ) is x < − 7 .
Explanation
Simplifying the Inequality First, let's simplify the given inequality: 4 − ( x + 2 ) < − 3 ( x + 4 ) . We need to isolate x to find the solution.
Distributing Terms Distribute the terms on both sides of the inequality: 4 − x − 2 < − 3 x − 12
Combining Like Terms Combine like terms on the left side: 2 − x < − 3 x − 12
Adding 3x to Both Sides Add 3 x to both sides of the inequality: 2 − x + 3 x < − 3 x − 12 + 3 x 2 + 2 x < − 12
Subtracting 2 from Both Sides Subtract 2 from both sides of the inequality: 2 + 2 x − 2 < − 12 − 2 2 x < − 14
Dividing by 2 Divide both sides by 2: 2 2 x < 2 − 14 x < − 7
Final Solution So the solution to the inequality is x < − 7 .
Examples
Understanding inequalities is crucial in many real-world scenarios. For example, when budgeting, you might need to ensure that your expenses are less than your income. If your income is represented by a constant and your expenses depend on a variable (like the number of items you buy), you can use inequalities to determine how many items you can afford while staying within your budget. Similarly, in science, inequalities can help define the range of acceptable values for experimental parameters to ensure a reaction occurs safely and effectively. Inequalities are also used in optimization problems, such as maximizing profit or minimizing cost, subject to certain constraints.
To solve the inequality 4 − ( x + 2 ) < − 3 ( x + 4 ) , we simplify it to x < − 7 . This indicates that any value of x less than -7 will satisfy the inequality.
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