Problem 42 - Algebra 2

Question

Solve each inequality. Then graph the solution set on a number line. \(\frac{1-4 p}{5}<0.2\)

Step-by-Step Solution

Answer

The solution is \(p > 0\). Graph an open circle at 0 and shade to the right.

Step 1: Eliminate the Fraction

To eliminate the fraction, multiply both sides of the inequality by 5 to cancel out the denominator.\[ \frac{1 - 4p}{5} < 0.2 \] becomes \[ 1 - 4p < 1 \] after multiplying each side by 5.

Step 2: Isolate the term with the variable

To isolate the term with the variable, subtract 1 from both sides of the inequality.\[ 1 - 4p - 1 < 1 - 1 \] simplifies to \[ -4p < 0 \]

Step 3: Solve for the variable

To solve for \(p\), divide both sides of the inequality by -4. Remember that dividing or multiplying both sides of an inequality by a negative number reverses the inequality sign.\[ p > 0 \] after solving the inequality.

Step 4: Graph the solution set on a number line

Since the solution is \(p > 0\), plot a number line with an open circle at 0 (since 0 is not included in the solution) and shade everything to the right of 0 to indicate all values greater than 0 are included.

Key Concepts

Solving Inequalities

Solving inequalities is a crucial skill in algebra that helps us understand relationships between different mathematical expressions. Unlike equations, where we find an exact value, inequalities tell us a range of values that satisfy a particular condition.
For example, solving the inequality \( \frac{1 - 4p}{5} < 0.2 \) involves finding the set of values for \( p \) that make the statement true. We approach this by:

• Ensuring each step maintains the inequality.
• Using operations that mirror those in equation solving, like addition or subtraction, to simplify expressions.
• Reversing the inequality sign when multiplying or dividing by a negative number.

This process allows us to find a solution that can be represented in a continuous range rather than a single number.

Graphing on a Number Line

Once we have a solution to our inequality, portraying it on a number line provides a visual representation. The solution \( p > 0 \) indicates that \( p \) can be any number greater than 0.
To graph this:

• Start by drawing a horizontal line, which represents possible values of \( p \).
• Mark the initial point (0 in this case) with an open circle to show that this point is not included in the solution.
• Shade the area to the right of the open circle to illustrate that all numbers greater than 0 satisfy the inequality.

Graphing helps in visually understanding the scope of solutions and is a helpful tool in both simple and complex inequalities.

Isolating Variables

Isolating the variable is a fundamental step in solving inequalities. This process involves manipulating the inequality so that the variable in question (here, \( p \)) stands alone on one side.
For the inequality \( 1 - 4p < 1 \):

• Subtract 1 from both sides to make the inequality \( -4p < 0 \).
• The goal is to make \( p \) isolated, so further we will divide by -4.
• Remember that dividing by a negative flips the inequality, giving \( p > 0 \).

This step-by-step isolation provides clarity and simplifies the process, making it easier to directly identify the range of values that meet the condition set by the inequality.

Fraction Elimination

Fractions can complicate inequalities, but they can be dealt with by removing the denominator. In the inequality \( \frac{1-4p}{5} < 0.2 \), we eliminate the fraction to simplify the inequality.
Here's how to do it:

• Multiply each side of the inequality by the denominator (5 in this case) to cancel it out.
• This operation results in a simpler form \( 1 - 4p < 1 \), making the inequality easier to handle.

Eliminating fractions not only simplifies the inequality but also aids in focusing on isolating the variable next. This makes the overall solving process more straightforward and manageable.