Solve the following inequality:
[tex]7 \leq 9 x-1.6\left(6 x^{\star}-4\right)[/tex]
Answer (2)
Distribute -1.6 into the parenthesis: 7 ≤ 9 x − 1.6 ( 6 x − 4 ) becomes 7 ≤ 9 x − 9.6 x + 6.4 .
Combine like terms: 7 ≤ − 0.6 x + 6.4 .
Subtract 6.4 from both sides: 0.6 ≤ − 0.6 x .
Divide by -0.6 and flip the inequality sign: x ≤ − 1 . The solution is x ≤ − 1 .
Explanation
Understanding the Inequality We are given the inequality 7 ≤ 9 x − 1.6 ( 6 x − 4 ) . Our goal is to isolate x to find the solution set.
Distributing First, distribute the − 1.6 into the parenthesis: 7 ≤ 9 x − 1.6 c d o t 6 x + 1.6 c d o t 4
Simplifying Simplify the inequality. We know that 1.6 c d o t 6 = 9.6 and 1.6 c d o t 4 = 6.4 , so we have: 7 ≤ 9 x − 9.6 x + 6.4
Combining Like Terms Combine like terms: 9 x − 9.6 x = − 0.6 x , so the inequality becomes: 7 ≤ − 0.6 x + 6.4
Isolating x Term Subtract 6.4 from both sides of the inequality: 7 − 6.4 ≤ − 0.6 x 0.6 ≤ − 0.6 x
Dividing by a Negative Number Divide both sides by -0.6. Remember to flip the inequality sign since we are dividing by a negative number: − 0.6 0.6 g e q x − 1 g e q x
Final Solution Therefore, the solution to the inequality is x ≤ − 1 .
Examples
Understanding inequalities is crucial in many real-world scenarios, such as budgeting. For instance, if you have a fixed budget and certain expenses, inequalities can help you determine how much you can spend on other items while staying within your budget. Similarly, in business, inequalities are used to model constraints like resource availability and production capacity to optimize profits. This problem demonstrates how to manipulate and solve inequalities, a fundamental skill for making informed decisions in various practical situations.
The solution to the inequality 7 ≤ 9 x − 1.6 ( 6 x − 4 ) is x ≤ − 1 . By distributing, combining like terms, and isolating x, we arrive at this conclusion. The final form shows all valid values of x that satisfy the condition are those that are less than or equal to -1.
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