Problem 8 - Algebra 2

Question

Solve each inequality. $$ \sqrt{2 x+3}-4 \leq 5 $$

Step-by-Step Solution

Answer

The solution is \(-\frac{3}{2} \leq x \leq 39\).

Step 1: Isolate the Square Root

First, we want to get the square root on one side of the inequality. Add 4 to both sides to get: \( \sqrt{2x+3} \leq 9 \).

Step 2: Remove the Square Root

To eliminate the square root, square both sides of the inequality. This will give us: \( 2x + 3 \leq 81 \) (since \( 9^2 = 81 \)).

Step 3: Solve for x

Subtract 3 from both sides to find \( 2x \leq 78 \). Then, divide each side by 2 to isolate \( x \), which results in \( x \leq 39 \).

Step 4: Verify the Domain

Since \( \sqrt{2x+3} \) is defined for non-negative values, check \( 2x+3 \geq 0 \). Solving this gives \( x \geq -\frac{3}{2} \). Thus, \( x \) must satisfy \( -\frac{3}{2} \leq x \leq 39 \).

Key Concepts

Understanding the Square Root

The concept of the square root is fundamental in solving inequalities like the one presented here. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 times 3 equals 9. In the given inequality, \( \sqrt{2x + 3} \) represents the square root of the expression \( 2x + 3 \).
Solving an inequality involving a square root often requires isolating the square root term first. In this exercise, we initially had \( \sqrt{2x + 3} - 4 \leq 5 \). By adding 4 to both sides, we isolate the square root: \( \sqrt{2x + 3} \leq 9 \).
This step is crucial, as it simplifies the inequality and sets us up for the subsequent actions needed to find a solution. You can think of it as "undoing" the operations that are complicating the variable's expression.

Tackling Inequality Solutions

Inequality solutions are about finding all possible values that satisfy a condition. Unlike equations that have precise solutions, inequalities can have a range of solutions. For example, solutions to the inequality \( x \leq 39 \) are the values of \( x \) that are less than or equal to 39.

• Inequalities can be solved by performing the same operations on both sides, just like equations.
• However, remember, multiplying or dividing by a negative number flips the inequality sign.

In our original exercise, by squaring both sides of \( \sqrt{2x + 3} \leq 9 \), we obtained \( 2x + 3 \leq 81 \). This transformation is valid because squaring is a monotonically increasing function for non-negative numbers, meaning it preserves the direction of the inequality.

Algebraic Manipulation Essentials

Algebraic manipulation is the tool we use to simplify the expressions and solve for the unknown variable, \( x \). It involves moving terms and changing the form of equations to reveal the variable's value.

• The basic operations we use include addition, subtraction, multiplication, and division.
• We aim to isolate the variable on one side of the inequality.

To solve the inequality \( 2x + 3 \leq 81 \), we first subtract 3 from both sides, leading to \( 2x \leq 78 \). Next, we divide each side by 2 to solve for \( x \), resulting in \( x \leq 39 \).
The final critical step is ensuring our solution respects the original domain of our problem. We solve \( 2x + 3 \geq 0 \) to maintain the domain where the original square root is defined, leading to \( x \geq -\frac{3}{2} \). Together, this yields the satisfied range: \( -\frac{3}{2} \leq x \leq 39 \). This consideration showcases the essence of algebraic manipulation in solving inequalities effectively.