Problem 32 - Algebra 2

Question

Solve each inequality. $$ \sqrt{3 x+6}+2 \leq 5 $$

Step-by-Step Solution

Answer

The solution is \(-2 \leq x \leq 1\).

Step 1: Isolate the Square Root

To solve the inequality \( \sqrt{3x + 6} + 2 \leq 5 \), first isolate the square root by subtracting 2 from both sides: \[ \sqrt{3x + 6} \leq 3 \]

Step 2: Eliminate the Square Root

Square both sides of the inequality to eliminate the square root. This gives us:\[ 3x + 6 \leq 9 \]

Step 3: Solve for x

Subtract 6 from both sides to isolate the \( x \) term:\[ 3x \leq 3 \]Now, divide both sides by 3 to solve for \( x \):\[ x \leq 1 \]

Step 4: Verify Domain Restrictions

Since square roots are only defined for non-negative values, ensure the expression inside the square root is non-negative:\[ 3x + 6 \geq 0 \]Solve for \( x \): \[ x \geq -2 \]

Key Concepts

Square Roots in Inequalities

Square roots can sometimes seem tricky, especially when they appear in inequalities. It is important to isolate the square root term when solving these expressions. This step helps simplify the problem and makes it easier to move forward with the solution.

• First, identify the square root term in the inequality.
• Isolate this term by performing the same operation on both sides of the inequality to maintain equality.
• Remember, square roots imply that the expression inside must be non-negative.

In our example,\[ \sqrt{3x + 6} + 2 \leq 5 \],we first subtract 2 from both sides to arrive at\[ \sqrt{3x + 6} \leq 3 \].This isolation makes it easier to eliminate the square root through the next step - squaring both sides.

Understanding Algebraic Expressions

Algebraic expressions, like \(3x + 6\) in our inequality, are essential elements in algebra used to describe mathematical relationships. They involve numbers, variables, and operation symbols, and they can be manipulated to simplify or solve equations and inequalities.

• Focus on each part of the algebraic expression one step at a time.
• Use operations like addition, subtraction, multiplication, and division to simplify the expression.
• Make sure any operation applied to solve an expression is balanced across the entire equation or inequality.

In this case, once we eliminated the square root, the expression became\[3x + 6 \leq 9\]. By subtracting 6 from both sides, the inequality simplifies to\[3x \leq 3\]. Dividing both sides by 3 gives\[x \leq 1\]. This step-by-step breakdown showcases the power of managing algebraic expressions effectively.

Domain Restrictions in Square Roots

Domain restrictions are important because they specify the values for which a mathematical expression is defined. For square roots, the expression inside must always be greater than or equal to zero to be valid, as negative values don't have real-number square roots in standard algebra.

• Identify the domain restrictions by setting the inside of the square root \(\geq 0\).
• Ensure that the solution to the inequality respects these domain restrictions.
• The solution must lie within this range to be considered valid.

In our example,\[3x + 6\geq 0\],solving gives us \[x \geq -2\]. Thus, when we solved the inequality \[x \leq 1\],it is important to note that \(x\) must also satisfy \(x \geq -2\). The final solution thus lies within the range\[-2 \leq x \leq 1\]. By checking domain restrictions, we guarantee that our solution is valid under real-number constraints.