Question

practice and problem - solving exercises
practice
determine whether the ordered pair is a solution of the linear inequality. see problem 1.

1. $y\leq - 2x + 1;(2,2)$
2. $x < 2;(-1,0)$
3. $y\geq 3x - 2;(0,0)$
4. $y > x - 1;(0,1)$
5. $y\geq -\frac{2}{5}x + 4;(0,0)$
6. $3y > 5x - 12;(-6,1)$

graph each linear inequality. see problem 2.

1. $y\leq x - 1$
2. $y\geq 3x - 2$
3. $y < - 4x - 1$
4. $y > 2x - 6$
5. $y < 5x - 5$
6. $y\leq \frac{1}{2}x - 3$
7. $y > - 3x$
8. $y\geq - x$

poweralgebra.com lesson 6 - 3 linear inequalities 397

Explanation:

First Section: Determine if ordered pairs are solutions

For each problem, substitute the ordered pair $(x,y)$ into the inequality and check if the resulting statement is true.
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Problem 8:

Step1: Substitute $x=2,y=2$

$2 \leq -2(2) + 1$

Step2: Simplify right-hand side

$2 \leq -4 + 1 \implies 2 \leq -3$

Step3: Verify truth value

This statement is false.

Problem 9:

Step1: Substitute $x=-1$

$-1 < 2$

Step2: Verify truth value

This statement is true.

Problem 10:

Step1: Substitute $x=0,y=0$

$0 \geq 3(0) - 2$

Step2: Simplify right-hand side

$0 \geq -2$

Step3: Verify truth value

This statement is true.

Problem 11:

Step1: Substitute $x=0,y=1$

$1 > 0 - 1$

Step2: Simplify right-hand side

$1 > -1$

Step3: Verify truth value

This statement is true.

Problem 12:

Step1: Substitute $x=0,y=0$

$0 \geq -\frac{2}{5}(0) + 4$

Step2: Simplify right-hand side

$0 \geq 4$

Step3: Verify truth value

This statement is false.

Problem 13:

Step1: Substitute $x=-6,y=1$

$3(1) > 5(-6) - 12$

Step2: Simplify both sides

$3 > -30 - 12 \implies 3 > -42$

Step3: Verify truth value

This statement is true.
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For each inequality:

1. First graph the boundary line: use a solid line for $\leq, \geq$ (includes the line) and a dashed line for $<, >$ (excludes the line).
2. Test a point not on the line (e.g., (0,0) if it is not on the boundary) to determine which half-plane to shade: if the point satisfies the inequality, shade its half-plane; if not, shade the opposite half-plane.

Answer:

(First Section):

1. No, (2,2) is not a solution.
2. Yes, (-1,0) is a solution.
3. Yes, (0,0) is a solution.
4. Yes, (0,1) is a solution.
5. No, (0,0) is not a solution.
6. Yes, (-6,1) is a solution.

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Second Section: Graphing Linear Inequalities