Question

In Exercises 37–44, solve the inequality. $$ -0.25 \sqrt{x}-6 \leq-3 $$

Short answer

The solution to the inequality -0.25 \(\sqrt{x}\) - 6 ≤ -3 is \(x \) ≥ 144.

Step-by-step solution

Outlining the Inequality

The initial inequality is -0.25 \(\sqrt{x}\) - 6 ≤ -3. The first step is to isolate the square root term. To do that, add 6 to both sides which turns the inequality into -0.25 \(\sqrt{x}\) ≤ 3.

Isolating the square root

The next step is to remove -0.25. To do that, you divide each side by -0.25. Since you're dividing by a negative number, you must reverse the comparison operator. This results in inequality \(\sqrt{x}\) ≥ -12.

Squaring both sides

Because we have a square root in our inequality, to simplify it we can square both sides. That results in \(x \) ≥ 144.

Define the solution set

The solution set to any square root inequality must satisfy non-negative real number interval [0, +∞). Therefore, in this case, the solution set is in the intersection of the two intervals [0, +∞) and [144, +∞). That gives us the final solution set [144, +∞), that means \(x \) ≥ 144.

Key concepts

Square Root Inequality

Square root inequalities involve expressions with square root symbols, and these can look a bit intimidating at first. Let's break it down simply!
In a square root inequality, like our example, the term involves a square root (\(\sqrt{x}\) in this case). These inequalities follow rules similar to normal inequalities but with a bit of a twist due to the square root operation.
The key challenge with square root inequalities is to deal carefully with the square root itself. This usually involves isolating the square root term first, before considering further steps like squaring both sides. Remember, any operations we perform should not violate the original inequality!

Solution Set

When solving inequalities, especially ones with square roots, finding the solution set is crucial. Think of the solution set as a range that includes all possible values that satisfy the inequality.
After simplifying an inequality, the final numerical range or interval is checked for compatibility with the square root’s requirement (only non-negative values work).
In our problem, after resolving the inequality, we have the inequality \(x \geq 144\). Thus, our solution set is \([144, +\infty)\). This solution set means any value of \(x\) starting from 144 and moving up to infinity solves the inequality.

Isolating Variables

Isolating the variable in inequality problems is all about "freeing" \(x\) from any encumbering numbers or operations. It involves algebraically rearranging the equation so that the variable stands alone on one side.
In our example, the goal is to isolate \(\sqrt{x}\). Initially, the square root is attached to other constants and coefficients, so we handle these step by step:

• Add 6 to both sides to annul the -6.
• Then, divide by -0.25 to further isolate \(\sqrt{x}\).

Remember, since dividing by a negative number flips the inequality sign, it’s vital to pay attention to this shift. Thus, every move should be done with the adjustment in mind to maintain the inequality.

Squaring Both Sides

When the variable inside a square root is isolated, sometimes the best way to simplify further is by squaring both sides. Squaring helps to eliminate the square root, giving a simpler form of the inequality.
In our exercise, we have \(\sqrt{x} \geq -12\), which at first might look off due to negative numbers involved in square root contexts. But here, technically it simplifies further to \(x \geq 144\) because squaring both values transforms the root terms accurately as long as following non-negative interval conditions are respected.
It's important to verify this step under the condition that all operations preserve the integrity of the inequality throughout the process."}]} ủy niệm ai ilottingag ell haeg Chris, but chest wipnotes robieren band elected midst.gov