Question

Solve the inequality. Then graph the solution set on the real number line. \((x+2)^{2}<25\)

Short answer

The solution to the inequality \((x+2)^2 < 25\) is -7 < x < 3. This represents all the real numbers that lies between -7 and 3.

Step-by-step solution

Simplify the inequality

The first step involves simplifying the inequality given to us. We know that \( (x+2)^2 < 25 \) simply means \( -25 < (x+2)^2 < 25 \).

Solve for x

We split the inequality into two parts and solve for x. This gives us the two following inequalities: \( -25 < (x+2)^2 \) and \( (x+2)^2 < 25 \). Solving these two inequalities gives us, \( -5 < x+2 < 5 \). Simplifying further, we find \( -7 < x < 3 \).

Graph the solution

The final step is to graph these results on the number line. The values for x are all the real numbers that lies between -7 and 3. Therefore, we draw an open circle at -7 and 3 (to represent that these numbers are not included in the solution set) and a line joining them.

Key concepts

Quadratic Inequalities

Quadratic inequalities are mathematical expressions where a quadratic expression is compared to another value using inequality symbols such as <, >, ≤, or ≥.

In the given exercise, we encounter the quadratic inequality \((x+2)^2 < 25\). This represents all the values of x for which the square of (x + 2) is less than 25. Unlike quadratic equations, which have exact solutions, inequalities show a range of solutions. To solve these inequalities, we often isolate the variable on one side and take square roots, keeping in mind that this may result in a positive or negative solution. Thus, the simplification process is vital for finding the correct interval of values for x that satisfy the inequality.

Inequality Simplification

Inequality simplification is crucial to solving quadratic inequalities as it allows us to reduce complex expressions to a simpler form where the solutions are more apparent.

The step-by-step solution of our exercise begins by rewriting the given inequality to identify the range in which the solutions lie, explicitly acknowledging the 'less than' aspect: \((x+2)^2 < 25\) represents the range of values for X we are looking for. When we solve this further, we find a simplified form, \(-7 < x < 3\), showing a clear interval of values that satisfy the original inequality. Simplification makes the final range of solutions easier to interpret and graph.

Graphing Solutions

Graphing solutions of inequalities on a number line helps to visually present the range of values that satisfy the inequality. When dealing with quadratic inequalities, such as in our exercise, the graph demonstrates that our variable x must lie within a certain interval for the inequality to hold true.

In practice, we place open circles at the boundary points to indicate that these numbers aren't included in the solution if the inequality is strict (with a < or > sign). If the inequality includes equality (with ≤ or ≥), we use closed circles instead. A line between these points shows the continuum of numbers that make the inequality true. The graph conveys this information intuitively.

Number Line Representations

A number line is a visual representation where each point corresponds to a real number, providing an invaluable tool for graphing the solution sets of inequalities.

In our exercise \(-7 < x < 3\), the number line representation consists of an open circle at -7 and another at 3, with a line segment connecting them. The open circles indicate that while -7 and 3 themselves are not solutions, all the real numbers that lie between them are solutions to the inequality. This visual approach on a number line offers students a clear understanding of the range of values that satisfy the quadratic inequality.