Problem 65 - Elementary Algebra Within Reach

Question

Solve and graph the inequality. $$3(x+1) \geq 2(x+5)$$

Step-by-Step Solution

Answer

The solution to the inequality is \(x \geq 7\). The graph will be a closed circle at \(x =7\) on a number line, with a line extending to the right to denote values greater than 7.

Step 1: Simplify Both Sides

Expand both sides of the inequality to simplify it. Therefore, the inequality becomes \(3x + 3 \geq 2x + 10\).

Step 2: Solve For \(x\)

Subtract \(2x\) from both sides of the inequality, we get \(x+3 \geq 10\). Then, subtract 3 from both sides, we get \(x \geq 7\). Thus, solution to the inequality is \(x \geq 7\).

Step 3: Graphing the Solution

On a number line, an open circle is put on number 7 to denote \(x>7\) and a closed circle to denote \(x = 7\). Drawing a line to the right of the number 7 suggests that the solutions for \(x\) are all the numbers greater than or equal to 7.

Key Concepts

Solving Inequalities

Solving inequalities is similar to solving regular equations, but with one key difference: the inequality symbol (\(<, >, \leq, \geq\)) must be preserved throughout the solution process. Here's how you can solve an inequality step by step:

• First, simplify both sides of the inequality just as you would do with an equation. Combine like terms or expand expressions where necessary.
• Next, use algebraic operations such as addition, subtraction, multiplication, or division to isolate the variable on one side of the inequality. However, remember that if you multiply or divide by a negative number, the inequality symbol must be reversed.
• The solution set you get represents all possible values that satisfy the inequality.

For example, consider the inequality we have: \[3(x+1) \geq 2(x+5)\]Simplification gives \[3x + 3 \geq 2x + 10\]Moving terms around, we solve and find \[x \geq 7\]. It means any number greater than or equal to 7 satisfies the inequality.

Graphing Inequalities

Graphing inequalities on a number line helps visually represent the solution set. It’s a great way to see which values satisfy the given inequality. Here’s how you can graph an inequality:

• Start by solving the inequality, as shown before. For example, we found that \(x \geq 7\).
• Draw a number line, and mark the critical point (in this case, \(7\)).
• Use a closed circle at 7 to indicate that 7 is part of the solution (\(x = 7\)). If the inequality was strict like \(x > 7\), you'd use an open circle.
• Draw a line or an arrow extending to the right from the marked point to show that the solution includes all numbers greater than or equal to 7.

Graphing helps learners better understand and verify the solution range by showing all possible solutions visually.

Linear Inequalities

Linear inequalities have a linear expression on one or both sides of the inequality symbol. These can be solved using similar techniques to linear equations, thanks to their straightforward nature. Here's a brief overview:

• Linear inequalities often appear as simple expressions like \(ax+b > c\) or \(ax+b \geq cx+d\).
• To solve, you simplify both sides and aim to isolate the variable, much like you would with a linear equation.”
• The primary difference is handling the inequality sign carefully, especially when multiplying or dividing by negatives.

The inequality we started with, \[3(x+1) \geq 2(x+5)\], exemplifies a linear inequality. Simplifying and moving terms gives you results that characterize the nature of inequalities, such as \(x \geq 7\), ensuring you're working with relationships rather than exact values. Unlike equations, inequalities broaden the scope by showing a range rather than just singular solutions.