Problem 39 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the inequality and sketch the solution on the real number line. Use a graphing utility to verify your solutions graphically. $$|5 x|>10$$

Step-by-Step Solution

Answer

The solution to the inequality \(|5x| > 10\) is \(x <-2\) and \(x > 2\)

Step 1: Divide the inequality by 5

In order to isolate \(x\), we must first get rid of the 5 that is multiplied to \(x\). This can be done by dividing all terms of the inequality by 5, giving us \( |x| > 2 \)

Step 2: Break absolute value into individual inequality

An absolute value inequality \(|x| > a\) breaks into two inequalities, \(x > a\) or \(x < -a\). Accordingly, break \(|x| > 2\) into \(x>2\) or \(x<-2\).

Step 3: Sketch on the real number line

The two separate inequalities indicate that the solution to \(x\) can be greater than 2 or lesser than -2. On the real number line, plot all values of \(x\) greater than 2 in one direction and all values less than -2 in the opposite direction. Excluding -2 and 2, as these aren't part of the solution due to the 'greater than' symbol. The line should look something like this: <-- -2 __ 2 -->

Step 4: Verify graphically

Plot the function \(y = |5x|\) as well as \(y=10\) using a graphing tool. Verify that \(y\) is greater than 10 only on intervals corresponding to \(x\) values less than -2 and greater than 2.

Key Concepts

Understanding Absolute Value Inequalities

Absolute value inequalities involve the absolute value expression, which essentially represents the distance of a number from zero on a number line, regardless of its direction. In this inequality, \(|5x|>10\), to make it simpler, we first divide both sides by 5 to result in \(|x|>2\). This means we are looking for all values whose distance from zero is greater than 2 on the number line. Since we are looking at inequalities, unlike an equation, it allows for a range of solutions instead of a single number.

Representing Inequalities on the Number Line

To represent solutions to inequalities such as \(x > 2\) or \(x < -2\), the number line is a helpful tool. On the number line:

• Locate points for \(2\) and \(-2\).
• Use open circles at these points to denote that 2 and -2 are not included in the solution (since \(|x| > 2\) does not include equality).
• Shade to the right of 2 to indicate numbers greater than 2, and shade to the left of -2 to indicate numbers less than -2.

This gives a visual indication of the set of x-values that satisfy the inequality \(|x|>2\). Ensuring you use open circles is key when the inequality symbol indicates greater than or less than, rather than equal to.

Graphical Verification of Solutions

Graphical verification is a technique used to check if the solutions derived from solving mathematical problems match when represented graphically. Using the graphing utility, plot the function \(y = |5x|\). Then add a horizontal line representing \(y = 10\). You should explore the points where the graph of \(y = |5x|\) lies above the line \(y = 10\), which corresponds with values for \(x < -2\) and \(x > 2\). By visually confirming the overlap, we verify that the solutions calculated previously accurately represent the inequality \(|5x|>10\). This not only strengthens understanding but also provides a clear visual proof.

Breaking Down Inequalities for a Clearer Solution

Breaking down inequalities involves separating the absolute value expression into more manageable parts. For the inequality \(|x| > 2\), this signals two possible regions of solution:

• One where \(x\) is simply greater than 2 (\(x > 2\)).
• Another where \(x\) is less than -2 (\(x < -2\)).

By splitting it into these two inequalities, we simplify the absolute value condition into more straightforward algebraic expressions. Each resulting inequality can be addressed individually, aiding clarity in finding the full set of solutions.