Solve the following inequality:
$-5 \leq 2-(-10 x+7)-9.7 x$
Answer (2)
Simplify the inequality by distributing and combining like terms: − 5 ≤ 2 − ( − 10 x + 7 ) − 9.7 x becomes − 5 ≤ − 5 + 0.3 x .
Isolate the x term by adding 5 to both sides: 0 ≤ 0.3 x .
Solve for x by dividing both sides by 0.3: 0 ≤ x .
The solution to the inequality is x ≥ 0 , which means x is greater than or equal to 0. x ≥ 0
Explanation
Understanding the Inequality We are given the inequality − 5 ≤ 2 − ( − 10 x + 7 ) − 9.7 x . Our goal is to solve for x , which means we want to isolate x on one side of the inequality to find the range of values that satisfy the inequality.
Distributing the Negative Sign First, let's simplify the right-hand side of the inequality by distributing the negative sign:
− 5 ≤ 2 + 10 x − 7 − 9.7 x
Combining Constant Terms Now, combine the constant terms on the right-hand side:
− 5 ≤ − 5 + 10 x − 9.7 x
Combining x Terms Next, combine the x terms on the right-hand side:
− 5 ≤ − 5 + 0.3 x
Isolating the x Term Now, we want to isolate the x term. Add 5 to both sides of the inequality:
− 5 + 5 ≤ − 5 + 5 + 0.3 x
0 ≤ 0.3 x
Solving for x Finally, divide both sides by 0.3 to solve for x :
0.3 0 ≤ 0.3 0.3 x
0 ≤ x
Expressing the Solution This means that x is greater than or equal to 0. In interval notation, this is written as [ 0 , ∞ ) .
Final Answer Therefore, the solution to the inequality is x ≥ 0 .
Examples
Understanding inequalities is crucial in many real-world scenarios. For example, when budgeting, you might want to ensure that your expenses are less than or equal to your income. If your income is represented by a constant and your expenses are a function of some variable (like time or usage), solving an inequality can help you determine how much you can spend without exceeding your income. Similarly, in physics, inequalities can be used to describe the range of possible values for variables like velocity or acceleration under certain constraints.
The solution to the inequality − 5 ≤ 2 − ( − 10 x + 7 ) − 9.7 x simplifies to x ≥ 0 . This means that x can be any value greater than or equal to 0. In interval notation, the solution is [ 0 , ∞ ) .
;
