Problem 30 - Intermediate Algebra

Question

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. \(\frac{2}{3} x+1>-9\) and \(\frac{3}{4} x-1>-10\)

Step-by-Step Solution

Answer

The solution is \( x > -12 \), graphed with an open circle on \(-12\), and in interval notation: \((-12, \infty)\).

Step 1: Solve the First Inequality

The first inequality is \( \frac{2}{3}x + 1 > -9 \). To isolate \( x \), first subtract 1 from both sides, resulting in \( \frac{2}{3}x > -10 \). Then, multiply each side by \( \frac{3}{2} \) to solve for \( x \): \( x > -15 \).

Step 2: Solve the Second Inequality

The second inequality is \( \frac{3}{4}x - 1 > -10 \). To isolate \( x \), add 1 to both sides, resulting in \( \frac{3}{4}x > -9 \). Then, multiply each side by \( \frac{4}{3} \) to solve for \( x \): \( x > -12 \).

Step 3: Combine the Solutions

Since we want the intersection (i.e., \( x \) must satisfy both inequalities), we take the larger of the two lower bounds: \( x > -12 \).

Step 4: Graph the Solution

On a number line, you would draw an open circle at \(-12\) and shade everything to the right, indicating all numbers greater than \(-12\).

Step 5: Express in Interval Notation

The interval notation for the solution \( x > -12 \) is \((-12, \infty)\).

Key Concepts

Interval Notation

Interval notation is a convenient way to express the set of solutions to inequalities. It uses brackets and parentheses to indicate which values are included and excluded in a range. In our compound inequality problem, we express solutions as open intervals or closed intervals. Open intervals, like

• \((-12, \infty)\)

demonstrate that the boundary value isn't part of the solution, indicated by parentheses. Here, \(-12\) is not included, denoting values greater than but not equal to \(-12\). Closed intervals would use square brackets,

• \([-12, \infty)\)

but they apply only when boundaries are included in the solution, such as \(x \geq -12\).
These concise notations help in conveying solutions of inequalities quickly and clearly. With only symbols and numbers, they precisely represent spans on the number line.

Solving Inequalities

Solving inequalities involves isolating the variable to find the range of values that satisfy the inequality. What we solve here are linear inequalities. The process generally involves simple algebraic steps:

• First, manipulate the inequality to isolate the variable term.
• Then, use operations such as addition, subtraction, multiplication, or division while keeping in mind that multiplying or dividing by a negative number reverses the inequality sign.

In our exercise, for

• \(\frac{2}{3}x + 1 > -9\)

we subtracted \(1\) from both sides and multiplied by \(\frac{3}{2}\) to finally get:

• \(x > -15\)

A similar process was applied to

• \(\frac{3}{4}x - 1 > -10\)

which resulted in

• \(x > -12\).

The solution to a compound inequality is the common portion meeting all conditions, which in our example is dependent on the stricter requirement \(x > -12\).
This guarantees the best solution satisfying both original equations.

Graphing Inequalities

Graphing inequalities involves illustrating the solution set on a number line, providing a visual representation of all possible solutions. For the compound inequality problem, our solution was \(x > -12\).
Here’s how you graph it:

• Draw a number line with an open circle at \(-12\) because \(-12\) is not included in the solution set.
• Shade the region to the right of \(-12\) towards positive infinity since the inequality represents all numbers greater than \(-12\).

This shaded region indicates solutions and shows that any point on this line can satisfy the inequality. Understanding how to graph inequalities is crucial as it visually showcases the feasible values of \(x\), confirming the algebraic solution. Graphing can clarify and reveal insights about inequalities, particularly useful in more complex scenarios.