Identifying and Comparing Information in Contextual Problems

In this module, you will identify solution methods for various algebraic problems and compare solutions methods. Engaging in this learning helps you see the “bigger picture” before you jump in to solve a problem. It also give you an opportunity to identify specific skills that are required to solve a problem and can help build your confidence as a problem-solver.

Let’s review your work from the previous lesson.

Identifying and Comparing Information in Contextual Problems

In the previous lesson, you practiced identifying important information in context and then you used that information to solve problems. Here is an example of those types of problems.

Worked Example:
Self Check:

Example

Recall that in the last lesson, we explored the water usage in various river basins.

Below is a table with a summary of population and water withdrawals for the six river basins in 2015.

Note the water is measured in acre-feet. An acre-foot is a unit of volume equal to about 1,233 meters cubed.

We calculated the percentage of population for several of the river basins and compared that percentage to the percentage of the water used by residents in those river basins. Here is an example from the previous lesson:

What percentage of the total population for the six river basins is in the Lower Colorado River Basin?

LaTeX: -10=x

Solution

To answer this question, we will create a ratio:

$LaTeX: \frac{PopulationofColoradoRiverBa\sin}{TotalPopulationofSixRiverBa\sin}$

$LaTeX: \frac{69090}{2099856}\approx0.0329$

Converting this value to a percentage, we have

$LaTeX: 0.0329\times100=3.29\%$

For the Texas Gulf Basin,

Part A: Calculate the percentage of the population in the river basin and compare that percentage to the percentage of the water used by residents in that river basin.

Part B: Compare this number to the percentage of the water used by the residents of that river basin.

Part A:

Part A:

$LaTeX: \frac{132284}{2099856}\approx0.063$

Converting to Percent: 6.3% of the population uses 11% of the water

Part B:

Part B: The population percentage is just a little less than double the population percentage.

Instructional Content:

Identifying Solution Methods for Algebraic Problems

When we are asked to solve an algebraic problem, we often start by analyzing the algebraic structure of the expressions in the problem and identifying a path for a solution method. Then we execute the method in a step-by-step manner.

Worked Example:

Use Algebraic methods to solve the following equation for x. Give an exact answer (no decimals).

$LaTeX: x^{\frac{3}{2}}=4$

When approaching this problem, we see that the variable is raised to a fractional exponent, and we recognize that we need to raise both sides of the equation to some power so that the power of the x term is 1. This means that we need to identify that power and then calculate the value of 4 raised to that power.

To identify the power, we need to consider the following:

$LaTeX: \left(x^{\frac{3}{2}}\right)^?=x^1$

So the ? should be $LaTeX: \frac{2}{3}$

$LaTeX: \left(x^{\frac{3}{2}}\right)^{\frac{2}{3}}=x^1$

So we have

$LaTeX: x=\left(4\right)^{\frac{2}{3}}$

Recall that when working with fractional exponents the denominator of the exponent describes the root as follows

$LaTeX: x=\left(4\right)^{\frac{2}{3}}=\left(\sqrt[3]{4}\right)^2$ or $LaTeX: x=\left(4\right)^{\frac{2}{3}}=\sqrt[3]{4^2}$

Finally, if we choose the second representation shown, we have

$LaTeX: x=\left(4\right)^{\frac{2}{3}}=\sqrt[3]{16}=2\sqrt[3]{2}$

Self Check:

Identify the solution methods for the following problem and then explain the step-by=step method for solving the problem as you solve the problem.

Solve for x: $LaTeX: \frac{7}{x+1}=2$

Solution

1. Multiply both sides by LaTeX: x+1 to eliminate the fraction:

$LaTeX: 7=2\left(x+1\right)$

2. Distribute the 2 on the right-hand side:

LaTeX: 7=2x+2

3. Subtract 2 from both sides to isolate the term with x:

LaTeX: 5=2x

4. Divide both sides by 2 to solve for x:

$LaTeX: x=\frac{5}{2}=2.5$

Comparing Solution Methods to Algebraic Problems

Citation: From NCDPI Math 2 Unpacking Document

Often there is more than one way to solve an algebraic problem. We will provide two solution methods for a problem and then compare those methods.

Worked Example:
Self Check:
Worked Example:
Self Check:

Worked Example

If you are asked to solve the equation:

LaTeX: 5x^2+10=90

You could use one of the methods shown below.

When you are choosing between these two methods, you may want to choose Method A since it does not require factoring but instead starts by collecting like terms and isolating the variable. This doesn’t mean that Method B is incorrect. Method A is a little more straight-forward.

Compare the following solution methods and identify the algebraic skills needed to complete the solution in each case.

Solve the equation for x: $LaTeX: \frac{3x}{4}=5-3x$

Method A:	Method B:
$LaTeX: \frac{3x}{4}=5-3x$	$LaTeX: \frac{3x}{4}=5-3x$
$LaTeX: 4\left(\frac{3x}{4}\right)=4\left(5-3x\right)$	$LaTeX: \frac{3x}{4}+3x=5$
	$LaTeX: \frac{3x}{4}+\frac{12}{4}x=5$
	$LaTeX: \frac{15x}{4}=5$
$LaTeX: x=\frac{20}{15}$	$LaTeX: \left(4\right)\frac{15x}{4}=\left(4\right)5$
$LaTeX: x=\frac{4}{3}$
	$LaTeX: x=\frac{20}{15}$ $LaTeX: x=\frac{4}{3}$

Solution:

Method A eliminates the denominator by multiplying both sides of the equation by 4. Then distributes and solves for x.

Method B combines the x-terms, but this requires finding a common denominator. Then eliminates the denominator and solves for x.

Method A seems more efficient because you don’t have to find a common denominator to combine the x terms.

Worked Example

Sometimes technology and graphical methods can be used to solve equations.

Compare the solution methods shown below for the equation:

$LaTeX: \sqrt{2x-5}=4$

Method A: Algebraic Method

1. Square both sides to eliminate the square root:

$LaTeX: \left(\sqrt{2x-5}\right)^2=4^2$

this simplifies to:

LaTeX: 2x-5=16

2. Add 5 to both sides to isolate 2x:

LaTeX: 2x=16+5

LaTeX: 2x=21

3. Divide both sides by 2 to solve for x:

$LaTeX: x=\frac{21}{2}=10.5$

Method B: Graphical Method

Graphing $LaTeX: y=\sqrt{2x+5}$ and LaTeX: y=4 , we can use Desmos to find the intersection point.

When comparing these methods, we might ask: What is our goal in solving the problem?

If we are learning and practicing algebraic skills, we would prefer Method A. If our goal is to understand the graphical representation of the solution, we would prefer to use Method B. If our goal is to solve the problem quickly, then using Method B would be preferable.

When you have completed the above lesson, click the next button in Canvas to progress.