Transcript solution of a linear inequality

6-6 Solving Linear Inequalities
A linear inequality is similar to a linear
equation, but the equal sign is replaced with
an inequality symbol.
A solution of a linear inequality is any
ordered pair that makes the inequality true.
6-6 Solving Linear Inequalities
Additional Example 1A: Identifying Solutions of
Inequalities
Tell whether the ordered pair is a solution of
the inequality.
(–2, 4); y < 2x + 1
y < 2x + 1
4 2(–2) + 1
4 –4 + 1
4 < –3 
(–2, 4) is not a solution.
Substitute (–2, 4) for (x, y).
6-6 Solving Linear Inequalities
Additional Example 1B: Identifying Solutions of
Inequalities
Tell whether the ordered pair is a solution of
the inequality.
(3, 1); y > x – 4
y>x−4
1
3–4
1> –1
Substitute (3, 1) for (x, y).

(3, 1) is a solution.
6-6 Solving Linear Inequalities
A linear inequality describes a region of a
coordinate plane called a half-plane.
All points in the half-plane region are solutions
of the linear inequality.
The boundary line of the region is the graph of
the related equation.
6-6 Solving Linear Inequalities
6-6 Solving Linear Inequalities
Graphing Linear Inequalities
Step 1
Solve the inequality for y (slopeintercept form).
Step 2
Graph the boundary line. Use a solid line
for ≤ or ≥. Use a dashed line for < or >.
Shade the half-plane above the line for y >
Step 3 or ≥. Shade the half-plane below the line
for y < or y ≤. Check your answer.
6-6 Solving Linear Inequalities
Example 2A: Graphing Linear Inequalities in Two
Variables
Graph the solutions of the linear inequality. Check
your answer.
y  2x – 3
Step 1 The inequality is
already solved for y.
Step 2 Graph the
boundary line y = 2x – 3.
Use a solid line for .
Step 3 The inequality is ,
so shade below the line.
6-6 Solving Linear Inequalities
Additional Example 2A Continued
Graph the solutions of the linear inequality.
Check your answer.
y  2x – 3
Check
y  2x – 3
0
2(0) – 3
0  –3 
Substitute (0, 0) for (x, y)
because it is not on the
boundary line.
A false statement means
that the half-plane
containing (0, 0) should
NOT be shaded. (0, 0) is
not one of the solutions,
so the graph is shaded
correctly.
6-6 Solving Linear Inequalities
Helpful Hint
Use the “test point” method shown in Example 2
to check your answers to linear inequalities. The
point (0, 0) is a good test point to use if it does
not lie on the boundary line. However, this
method will check only that your shading is
correct. It will not check the boundary line.
6-6 Solving Linear Inequalities
Example 2B: Graphing Linear Inequalities in Two
Variables
Graph the solutions of the linear inequality.
Check your answer.
5x + 2y > –8
Step 1 Solve the inequality for y.
5x + 2y > –8
–5x
–5x
2y > –5x – 8
y>
x–4
Step 2 Graph the boundary line y =
dashed line for >.
x – 4. Use a
6-6 Solving Linear Inequalities
Additional Example 2B Continued
Graph the solutions of the linear inequality.
Check your answer.
5x + 2y > –8
Step 3 The inequality is >, so
shade above the line.
6-6 Solving Linear Inequalities
Additional Example 2B Continued
Graph the solutions of the linear inequality.
Check your answer.
5x + 2y > –8
Check
Substitute ( 0, 0)
y>
x–4
for (x, y)
because it is
not on the
0
(0) – 4
boundary line.
0
–4
The point (0, 0)
0 > –4
satisfies the
inequality, so the
graph is correctly
shaded.
6-6 Solving Linear Inequalities
Example 2C: Graphing Linear Inequalities in two
Variables
Graph the solutions of the linear inequality.
Check your answer.
4x – y + 2 ≤ 0
Step 1 Solve the inequality for y.
4x – y + 2 ≤ 0
–y
–1
≤ –4x – 2
–1
y ≥ 4x + 2
Step 2 Graph the boundary line y = 4x + 2.
Use a solid line for ≥.
6-6 Solving Linear Inequalities
Additional Example 2C Continued
Graph the solutions of the linear inequality.
Check your answer.
Step 3 The inequality is ≥, so
shade above the line.
6-6 Solving Linear Inequalities
Additional Example 2C Continued
Check
y ≥ 4x + 2
3
4(–3)+ 2
3
–12 + 2
3 ≥ –10 
Substitute ( –3, 3) for (x, y) because
it is not on the boundary line.
The point (–3, 3) satisfies the inequality,
so the graph is correctly shaded.
6-6 Solving Linear Inequalities
Additional Example 4A: Writing an Inequality from a
Graph
Write an inequality to represent the graph.
y-intercept: 1; slope:
Write an equation in slopeintercept form.
The graph is shaded above a
dashed boundary line.
Replace = with > to write the inequality
6-6 Solving Linear Inequalities
Additional Example 3a: Application
Ada has at most 285 beads to make jewelry. A
necklace requires 40 beads, and a bracelet
requires 15 beads.
Let x represent the number of necklaces and y the
number of bracelets.
Write an inequality. Use ≤ for “at most.”
6-6 Solving Linear Inequalities
Additional Example 3a Continued
Necklace
beads
40x
plus
bracelet
beads
is at
most
285
beads.
+
15y
≤
285
Solve the inequality for y.
40x + 15y ≤ 285
–40x
–40x
Subtract 40x from
15y ≤ –40x + 285
both sides.
Divide both sides by 15.
6-6 Solving Linear Inequalities
Additional Example 3b
b. Graph the solutions.
Step 1 Since Ada cannot make a
negative amount of jewelry, the
system is graphed only in
Quadrant I. Graph the boundary
line
for ≤.
=
. Use a solid line
6-6 Solving Linear Inequalities
Additional Example 3b Continued
b. Graph the solutions.
Step 2 Shade below the line.
Ada can only make whole
numbers of jewelry. All points
on or below the line with
whole number coordinates are
the different combinations of
bracelets and necklaces that
Ada can make.
6-6 Solving Linear Inequalities
Additional Example 3c
c. Give two combinations of necklaces and
bracelets that Ada could make.
Two different combinations of
jewelry that Ada could make
with 285 beads could be 2
necklaces and 8 bracelets or 5
necklaces and 3 bracelets.
(2, 8)


(5, 3)