Problem 55 - Algebra and Trigonometry

Question

Find the solutions of the inequality by drawing appropriate graphs. State each answer correct to two decimals. $$ (x+1)^{2}<(x-1)^{2} $$

Step-by-Step Solution

Answer

The solution to the inequality is \(x < 0\).

Step 1: Expand the Expressions

Expand both sides of the inequality. For the left side: \((x+1)^2 = x^2 + 2x + 1\).For the right side: \((x-1)^2 = x^2 - 2x + 1\). So, the inequality becomes: \(x^2 + 2x + 1 < x^2 - 2x + 1\).

Step 2: Simplify the Inequality

Remove \(x^2 + 1\) from both sides. This gives: \(2x < -2x\). Add \(2x\) to both sides to get: \(4x < 0\).

Step 3: Solve for \(x\)

Divide both sides by 4: \(x < 0\). This represents the solution to the inequality.

Step 4: Verify with a Graph

Plot the graphs of \((x+1)^2\) and \((x-1)^2\). The inequality \((x+1)^2 < (x-1)^2\) holds wherever the graph of \((x+1)^2\) is below \((x-1)^2\). Observe that this occurs when \(x < 0\).

Key Concepts

Graphical Solution of Inequalities

Understanding quadratic inequalities can become more intuitive through graphical solutions. This involves plotting the graphs that represent both sides of the inequality equation. In this exercise, we graph

• \((x+1)^2\), a parabola shifted left by 1 unit from the origin,
• \((x-1)^2\), another parabola shifted right by 1 unit.

Both parabolas open upwards, but they are located symmetrically around the y-axis. To solve the inequality \((x+1)^2 < (x-1)^2\), we find the region where the first graph lies below the second graph. By plotting these, it's clear that the graph of \((x+1)^2\) falls below \((x-1)^2\) when \(x < 0\). This visual approach helps confirm and understand the algebraic solution.

Expanding Quadratic Expressions

Expanding quadratic expressions involves using the distributive property to rewrite a squared binomial as a trinomial. Consider the expression \((x+1)^2\). By applying

• the distributive property: \((x + 1)(x + 1)\),
• then distributing again: \(x(x+1) + 1(x+1)\),
• you get \(x^2 + x + x + 1\),
• which simplifies to \(x^2 + 2x + 1\).

Similarly, expanding \((x-1)^2\) results in \(x^2 - 2x + 1\). Expanding is a crucial step in solving inequalities as it often simplifies the inequality, enabling easier manipulation and comparison of both sides.

Simplifying Inequalities

Simplifying inequalities often involves combining like terms and eliminating unnecessary variables or constants. In the inequality

• \(x^2 + 2x + 1 < x^2 - 2x + 1\),

you can remove common terms \(x^2 + 1\) from both sides to simplify it effectively. This leaves \(2x < -2x\). To isolate the variable \(x\), we add \(2x\) to both sides, leading to \(4x < 0\).
The final step involves solving the simplified inequality by dividing each side by 4, producing \(x < 0\).
This simplification makes the underlying relationship between variables clearer, allowing you to reach a solution more efficiently. Simplifying transformations are particularly useful in complex inequalities where direct solutions are not obvious.