Problem 38 - Algebra 2

Question

Solve each inequality. $$ \sqrt{c+5}+\sqrt{c+10} > 2 $$

Step-by-Step Solution

Answer

The solution is \( c > -5 \).

Step 1: Isolate Square Roots

First, rearrange the inequality to separate terms with square roots on one side. This is already done in our inequality, so we can focus on the analysis as it is: \( \sqrt{c+5} + \sqrt{c+10} > 2 \).

Step 2: Determine Domain Constraints

For both \( \sqrt{c+5} \) and \( \sqrt{c+10} \) to be real, we need \( c+5 \geq 0 \) and \( c+10 \geq 0 \). This implies \( c \geq -5 \) and \( c \geq -10 \). Therefore, the relevant constraint is \( c \geq -5 \).

Step 3: Consider Critical Points

Let's consider when both square root terms could add up to just 2. If we hypothesize \( \sqrt{c+5} = 1 \) and \( \sqrt{c+10} = 1 \), this leads to solving \( c+5 = 1 \) and \( c+10 = 1 \). Solving them yields contradictions with the domain, so we focus instead on examining test intervals.

Step 4: Test Critical Values in Relation to Domain

To check when \( \sqrt{c+5} + \sqrt{c+10} \) exceeds 2, consider test points such as \( c = 0, 1, 2,...\) for simplicity and intuition within the domain. For instance, \( c = 0 \) gives \( \sqrt{5} + \sqrt{10} > 2 \). Since \( \sqrt{5} \approx 2.24 \) and \( \sqrt{10} \approx 3.16 \), this sum exceeds 2.

Step 5: Confirm General Trend

The inequality \( \sqrt{c+5} + \sqrt{c+10} > 2 \) persists for \( c \geq -5 \), as the value of each square root function increases with \( c \), and thus their sum is always greater than 2 for non-negative 'c' values. Therefore, every \( c \geq -5 \) satisfies the inequality.

Key Concepts

Square Roots

Square roots are mathematical functions that find the number, which when multiplied by itself, gives the original number. For example, the square root of 4 is 2 because \( 2 \times 2 = 4 \). Square roots can be tricky when dealing with inequalities, but the rules are straightforward:

• Square roots of non-negative numbers are always real.
• Negative numbers do not have real square roots. Instead, they have imaginary counterparts.

In our inequality \( \sqrt{c+5} + \sqrt{c+10} > 2 \), we have two expressions under the square root. It's important to ensure that these expressions, \( c+5 \) and \( c+10 \), are non-negative to keep the square roots real. This is a vital step in solving such inequalities.

Real Numbers

Real numbers include all numbers on the number line, encompassing both rational and irrational numbers. They are essential for solving inequalities involving square roots since the solutions must be real to make sense in a mathematical and real-world context.

• Real numbers can be positive, negative, or zero.
• They encompass integers, fractions, and non-repeating decimals.

In our exercise, the domain of \( c \) must ensure the expressions inside the square roots are non-negative, making \( c \) a real number greater than or equal to -5. By restricting \( c \) in such a way, we're specifying only the real solutions that satisfy the inequality.

Test Intervals

Test intervals are a method used to solve inequalities by breaking down possible solutions into manageable segments. You test values within each interval to see if they satisfy the inequality.

• Choose points within the domain that make calculations simple.
• Checking consecutive values can reveal a pattern or confirm the inequality is satisfied.

For example, by choosing \( c = 0 \), we calculate \( \sqrt{5} + \sqrt{10} \) and find it exceeds 2. Testing different values in intervals like \( 0, 1, \text{ and } 2 \) confirmed the inequality holds true. By validating this trend across the interval, we can assert the solution confidently.

Domain Constraints

Domain constraints are the permissible values for a variable that keep expressions within an inequality defined and real.

• They derive from conditions such as non-negativity under square roots or denominator restrictions in fractions.
• Determining domain constraints is crucial as it defines the scope within which the problem can be solved.

In our problem, since we have square roots \( \sqrt{c+5} \) and \( \sqrt{c+10} \), we set domain constraints \( c \geq -5 \) to ensure these square roots yield real numbers. Understanding domain constraints helps in simplifying and solving inequalities with precision.