Question

Use the graphical method to find the simultaneous solution set of $$ \begin{array}{r} x^{2}+x-2>0 \\ \text { and } 3 / 4 x+3 / 2<0 \end{array} $$

Short answer

The simultaneous solution set for the given inequalities is found by graphing both inequalities on a number line and finding their intersection. We determine that the first inequality is true when \(x < -2\) or \(x > 1\), and the second inequality is true when \(x < -2\). The intersection of the solution sets is when \(x < -2\), so the simultaneous solution set for the inequalities is \(x < -2\).

Step-by-step solution

Solve the first inequality for x

To solve the inequality \(x^2 + x - 2 > 0\), we first find its roots by setting the equation equal to zero: \(x^2 + x - 2 = 0\) To find the roots, we factor the equation: \((x - 1)(x + 2) = 0\) The roots are \(x = 1\) and \(x = -2\). With these roots, we can construct a sign table: | Interval | Test Point | Sign | |---------------|------------|---------| | \(-∞, -2\) | -3 | Positive | | \((-2, 1)\) | 0 | Negative | | \(1, ∞\) | 2 | Positive | So, the inequality \(x^2 + x - 2 > 0\) is true when \(x < -2\) or \(x > 1\).

Solve the second inequality for x

To solve the inequality \( \frac{3}{4}x + \frac{3}{2} < 0\), we first isolate x: \(\frac{3}{4}x < -\frac{3}{2}\) Now, divide both sides by \(\frac{3}{4}\): \(x < -2\) So, the inequality is true when \(x < -2\).

Graph the inequalities and find the simultaneous solution set

Now, we will graph both inequalities on a number line to find their simultaneous solution set. Since both inequalities are satisfied when \(x < -2\), that part of the number line is shaded for the intersection of the two inequalities. The other part, \(x > 1\), is true for the first inequality, but not for the second. Therefore, the solution set of both inequalities simultaneously is \(x < -2\).

Key concepts

Inequalities

Inequalities are mathematical expressions involving symbols that depict a range of values rather than a single solution, unlike equations. The most common inequality symbols are greater than (">"), less than ("<"), greater than or equal to ("\geq"), and less than or equal to ("\leq"). These expressions are foundational in determining the range within which a variable can exist.

In the given exercise, two inequalities are given:

• The first inequality is a quadratic inequality: \(x^2 + x - 2 > 0\).
• The second is a linear inequality: \(\frac{3}{4}x + \frac{3}{2} < 0\).

Solving inequalities involves finding the set of values for which the expression holds true. In equations, this process would end with a value or values that satisfy a balancing act on both sides. Here, however, we seek ranges."

Understanding inequalities and how to solve them prepares students for more advanced mathematical concepts. This provides a broader understanding of how variables behave within a given set of constraints.

Graphical Method

The graphical method is a visual approach to solving inequalities by drawing graphs that represent each condition and then identifying the overlap of these regions. The graphical method helps in visualizing which values satisfy two or more inequalities at the same time, or simultaneously.

In our example, both inequalities are solved separately on a line; then, the solutions are superimposed to find common values. The process involves the following steps:

• Graph each inequality on the same number line or coordinate plane.
• Identify the regions where the conditions of each inequality are true.
• Highlight the overlap of these regions, indicating where all inequalities are true simultaneously.

Using a number line for the graphical method is especially effective in single-variable inequalities, as it allows us to immediately see where both conditions intersect. For more complex, multi-variable systems, graphs on the coordinate plane might be more appropriate, using shaded areas to represent solution spaces.

Solution Sets

The solution set is a collection of all possible values that satisfy a given inequality or set of inequalities. It provides the answer by indicating the range of x-values (or y-values, z-values, etc., depending on the problem) that make the inequality true.

In this exercise, we combine the solutions from both inequalities to identify a range that satisfies them both:

• The first inequality \(x^2 + x - 2 > 0\) simplifies to the range \(x < -2\) or \(x > 1\).
• The second inequality \(\frac{3}{4}x + \frac{3}{2} < 0\) simplifies to \(x < -2\).

By finding where these ranges intersect, we determine that the simultaneous solution set is \(x < -2\). It is crucial to understand that only the overlapping parts of the solutions where both inequalities are true can be considered part of the solution set. This ensures that all conditions of the problem are accounted for.

Factoring Quadratics

Factoring quadratics is a technique used to simplify quadratic expressions, making them easier to solve or analyze. It involves rewriting a quadratic expression in the form of a product of its linear factors.

The expression for the first inequality, \(x^2 + x - 2 > 0\), can be factored into \((x - 1)(x + 2)\).

• This step is achieved by finding two numbers whose product equals the constant term (-2) and whose sum equals the linear coefficient (+1).
• Through trial or by deriving solutions based on arithmetic, the factors reveal the roots of the equation \(x = 1\) and \(x = -2\).

Understanding the principle of factoring is not only key to solving quadratic equations and inequalities but also fundamental in algebraic manipulation and graphing parabolas. By converting a quadratic equation into its factor form, students gain insight into its zeros or solutions, and the behavior of the graph of the quadratic, such as whether it opens upwards or downwards.