Solve the inequality $-24 \leq x-3-8 x$
A) $x \leq 3$
B) $x \geq-9$
C) $x \geq 3$
D) $x \geq-3$
Answer (1)
Combine like terms: − 24 ≤ x − 3 − 8 x becomes − 24 ≤ − 7 x − 3 .
Add 3 to both sides: − 21 ≤ − 7 x .
Divide by -7 and flip the inequality sign: 3 ≥ x .
Rewrite the inequality: x ≤ 3 .
Explanation
Understanding the Inequality We are given the inequality − 24 ≤ x − 3 − 8 x . Our goal is to isolate x and solve for it.
Combining Like Terms First, we simplify the right side of the inequality by combining like terms: x − 3 − 8 x = x − 8 x − 3 = − 7 x − 3
Rewriting the Inequality Now the inequality looks like this: − 24 ≤ − 7 x − 3
Isolating the x Term Next, we want to isolate the term with x . We add 3 to both sides of the inequality: − 24 + 3 ≤ − 7 x − 3 + 3
Simplifying This simplifies to: − 21 ≤ − 7 x
Dividing by -7 Now, we divide both sides by -7. Remember that when we divide by a negative number, we must flip the inequality sign: − 7 − 21 ≥ − 7 − 7 x
Simplifying This simplifies to: 3 ≥ x
Final Answer We can rewrite this as: x ≤ 3
Examples
Understanding inequalities is crucial in many real-world scenarios. For instance, when budgeting, you might want to ensure your expenses are less than or equal to your income. If your income is I and your expenses are E , the inequality E ≤ I must hold. Similarly, in engineering, inequalities are used to define safety margins. If a bridge can withstand a load of L , and the expected load is l , then l ≤ L must be true to ensure safety. This problem helps you practice the algebraic manipulation needed to solve such inequalities.
