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Topic 1 Lesson 5

Solving Equations and Inequalities by Graphing

I CAN…
use graphs and tables to approximate solutions to algebraic equations and
inequalities.

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Topic 1 Lesson 5

A homeowner has 32 feet of fencing to build three sides of a rectangular
chicken run.

A.

Make a table of values for the length, width, and area of different rectangular chicken
runs that will utilize 32 feet of fencing. Then write a function for the area, in terms of
width, of a rectangular run using this much fencing.

B.

Graph your function.

C.

Reason Explain what happens where the graph intersects the x-axis.

CRITIQUE & EXPLAIN

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Topic 1 Lesson 5

ESSENTIAL QUESTION

How can you solve an equation or inequality by graphing?

CONCEPTUAL UNDERSTANDING

Use a Graph to Solve an Equation

How can you use a graph to solve an equation?

A.

Solve −3x + 20 = 0.2x + 4 by graphing.

To solve an equation by graphing, write two new equations by setting y equal to each in the original
equation.

Graph the two equations and identify the points of intersection. The solutions are the x-values that
produce the same y-values for both expressions.

So −3x + 20 = 0.2x + 4 when x = 5.

EXAMPLE 1

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Topic 1 Lesson 5

Use a Graph to Solve an Equation

How can you use a graph to solve an equation?

B.

Solve by graphing.

The solutions to the equation are x = 2 and x = 10.

You can verify these values by substituting them back into the original equation.

EXAMPLE 1
CONCEPTUAL UNDERSTANDING

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Topic 1 Lesson 5

Use a Graph to Solve an Equation

Try It!

1.

Use a graph to solve the equation.

a.

5x − 12 = 3

b.

EXAMPLE 1

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Topic 1 Lesson 5

Solve a One-Variable Inequality by Graphing

How can you use a graph to solve an inequality?

A.

Solve x2 − 4 > 0.

To solve the inequality, identify the values of x that make the value of the expression x2 − 4 greater
than 0. Graph the equation y = x2 − 4 by translating the parent function y = x2 down 4 units.

The graph of the function is positive over the intervals (−∞, −2) and (2, ∞). So x2 − 4 > 0 when
x < −2 or x > 2.

EXAMPLE 2
APPLICATION

COMMON ERROR
You may think that you need to find interval(s) where x > 0. However, you need to find
where x2 − 4 > 0, so you are looking for interval(s) where y > 0.

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Topic 1 Lesson 5

Solve a One-Variable Inequality by Graphing

How can you use a graph to solve an inequality?

B.

A motorcycle is 40 mi ahead of a car. When will
the car be ahead of the motorcycle?

Let x represent the number of hours since the car started
traveling. The expression 60x represents the distance the
car travels in x hours. The expression 40x represents the
distance the motorcycle travels in x hours.
To solve, we need to determine when the number of miles
the car travels exceeds the number of miles the motorcycle
travels.

Solve 60x > 40x + 40.

Set y = 60x and y = 40x + 40, and then graph both equations.

EXAMPLE 2
APPLICATION

COMMON ERROR
Recall that the motorcycle started out 40 miles ahead of the car. Remember to add this
distance to the expression that represents the total distance the motorcycle has traveled.

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Topic 1 Lesson 5

Solve a One-Variable Inequality by Graphing

How can you use a graph to solve an inequality?

B.

A motorcycle is 40 mi ahead of a car. When will
the car be ahead of the motorcycle?

The car will be ahead of the motorcycle any time after 2 hours. You can verify the solution by
selecting any number greater than 2 and substituting it into the original inequality.

EXAMPLE 2
APPLICATION

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Topic 1 Lesson 5

Solve a One-Variable Inequality by Graphing

Try It!

2.

Use a graph to solve each inequality.

a.

x2 + 6x + 5 ≥ 0

b.

x + 3 > 7 − 3x

EXAMPLE 2

Multiple Choice

What is the color of an orange?

• purple
• orange
• yellow
• green

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Topic 1 Lesson 5

Use a Table to Solve an Equation

Use a graph and tables to solve the equation x2 − 4x + 1 = x − 2.

Sketch the graphs of y = x2 − 4x + 1 and y = x − 2. Identify the points of intersection to find initial
estimate(s) of the solution value(s).

Neither solution is easily read from the graph. You can use a table to get more accurate estimates for
the solutions.

EXAMPLE 3

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Topic 1 Lesson 5

Use a Table to Solve an Equation

Use a graph and tables to solve the equation x2 − 4x + 1 = x − 2.

One solution is approximately x ≈ 0.697.

You can use a similar method to approximate the second solution.

EXAMPLE 3

USE APPROPRIATE TOOLS
Create tables using a calculator
or spreadsheet. When you let
technology perform the
calculations, you can
concentrate on what the
numbers mean.

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Topic 1 Lesson 5

Use a Table to Solve an Equation

Try It!

3.

The equation x2 − 4x + 1 = x − 2 has a second solution in the interval 4 < x < 5. Use
a spreadsheet to approximate this solution to the nearest thousandth.

EXAMPLE 3

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Topic 1 Lesson 5

Use Graphing Technology to Solve Equations

A.

Use graphing technology to approximate the solutions of the equation
−x2 + 8x − 13 = |x − 4| to the nearest tenth.

Graph y = −x2 + 8x − 13 and y = |x − 4|.

Use the INTERSECT feature to find the approximate solutions.

The INTERSECT feature shows that the equation has solutions x ≈ 2.7 and x ≈ 5.3.

EXAMPLE 4

STUDY TIP
Test your approximate solutions in the original equation. The value of the expression
on the left side should be very close to the value of the expression on the right.

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Topic 1 Lesson 5

Use Graphing Technology to Solve Equations

Try It!

4.

Use graphing technology to approximate the solutions of the equation
x2 + 2x − 1 = |x + 2| + 2 to the nearest tenth.

EXAMPLE 4

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Topic 1 Lesson 5

CONCEPT SUMMARY

Solving Equations and Inequalities by Graphing

WORDS

Graph each function, and identify the x-coordinates of the points of intersection.

One solution is 0.
The second solution is −3.

TABLE

Graphs may not always yield integer results. Tables may be used to find solutions.

In general, graphing can be used to
solve equations by finding intersection
points. Graphing can also be used to solve
inequalities by finding the intersection
points, and then comparing the graphs.

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Topic 1 Lesson 5

Do You UNDERSTAND?

1.

ESSENTIAL QUESTION How can you solve an equation or inequality by graphing?

2.

Communicate Precisely What is an advantage of solving an equation graphically by
finding the points of intersection?

3.

Error Analysis Ben said the graph of the inequality −x2 + 9 > 0 shows the solution is
x < −3 or x > 3. Is Ben correct? Explain.

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Topic 1 Lesson 5

Do You KNOW HOW?

4.

Using the graph below, what is the solution to −2x + 4 = −2? How can you tell?