Asked by tomdaya
To earn points back for #1, webmail me an explanation on how you would graph an inequality. How do you know if the circle is open or closed? How do you know which direction the arrow should point?
For #2 and #3, explain the steps to solve the problem. Pay careful attention on #3 with the negative coefficient.
Answers
Answered by
GPT 3.5
To graph an inequality, you will need to follow a few steps:
Step 1: Start by writing down the inequality in the form "y > " or "y < " or "y ≥ " or "y ≤ " or "y ≠ " followed by the expression on the right side.
Step 2: Treat the inequality as an equation and graph the corresponding line or curve. If it is a less than or greater than inequality (without the equal sign), use a dashed line. If it is a less than or equal to or greater than or equal to inequality, use a solid line.
Step 3: Determine whether the inequality is less than (<) or greater than (>). If the inequality symbol is < or ≤, the arrow should point towards the left side of the graph. If the inequality symbol is > or ≥, the arrow should point towards the right side of the graph.
Step 4: Identify a test point within the shaded region, which will help determine which area represents the solutions. You can select any point, but it's often easiest to choose (0,0) or another point with small integer values.
Step 5: Substitute the test point's coordinates into the original inequality. If the inequality is true when you substitute the test point, shade the region that contains the test point. If it is false, shade the opposite region. This shading represents the solution set for the inequality.
For example, let's say you have the inequality y > 2x - 3. The steps to graph this inequality would be as follows:
Step 1: Write down the inequality as y > 2x - 3.
Step 2: Graph the line y = 2x - 3 using a dashed line since it is a greater than inequality.
Step 3: Since the inequality is greater than, the arrow should point towards the right side of the graph.
Step 4: Choose a test point, let's say (0,0). Substitute the coordinates into the inequality: 0 > 2(0) - 3. This simplifies to 0 > -3, which is true.
Step 5: Shade the region to the right of the line, as this is where the solutions lie.
Regarding #3 with the negative coefficient, the steps to solve the problem are as follows:
Step 1: Write down the inequality in the desired form. For example, if given -3x + 2 ≤ 5, you can rewrite it as -3x ≤ 5 - 2.
Step 2: Simplify the expression on the right side, -3x ≤ 3.
Step 3: Divide both sides of the inequality by the coefficient of x. In this case, divide by -3. However, when dividing or multiplying by a negative number, you must also flip the inequality symbol. So, the inequality becomes x ≥ -1.
Step 4: Graph the inequality x ≥ -1 by drawing a solid line at x = -1 and shading the region to the right of the line.
Step 5: Remember to choose a test point, let's say (0,0), and substitute it into the original inequality -3x + 2 ≤ 5. After simplifying, you should get a true statement, confirming that the shaded region is the solution.
I hope this explanation helps you understand how to graph an inequality and solve problems involving negative coefficients. If you have any further questions, feel free to ask!
Step 1: Start by writing down the inequality in the form "y > " or "y < " or "y ≥ " or "y ≤ " or "y ≠ " followed by the expression on the right side.
Step 2: Treat the inequality as an equation and graph the corresponding line or curve. If it is a less than or greater than inequality (without the equal sign), use a dashed line. If it is a less than or equal to or greater than or equal to inequality, use a solid line.
Step 3: Determine whether the inequality is less than (<) or greater than (>). If the inequality symbol is < or ≤, the arrow should point towards the left side of the graph. If the inequality symbol is > or ≥, the arrow should point towards the right side of the graph.
Step 4: Identify a test point within the shaded region, which will help determine which area represents the solutions. You can select any point, but it's often easiest to choose (0,0) or another point with small integer values.
Step 5: Substitute the test point's coordinates into the original inequality. If the inequality is true when you substitute the test point, shade the region that contains the test point. If it is false, shade the opposite region. This shading represents the solution set for the inequality.
For example, let's say you have the inequality y > 2x - 3. The steps to graph this inequality would be as follows:
Step 1: Write down the inequality as y > 2x - 3.
Step 2: Graph the line y = 2x - 3 using a dashed line since it is a greater than inequality.
Step 3: Since the inequality is greater than, the arrow should point towards the right side of the graph.
Step 4: Choose a test point, let's say (0,0). Substitute the coordinates into the inequality: 0 > 2(0) - 3. This simplifies to 0 > -3, which is true.
Step 5: Shade the region to the right of the line, as this is where the solutions lie.
Regarding #3 with the negative coefficient, the steps to solve the problem are as follows:
Step 1: Write down the inequality in the desired form. For example, if given -3x + 2 ≤ 5, you can rewrite it as -3x ≤ 5 - 2.
Step 2: Simplify the expression on the right side, -3x ≤ 3.
Step 3: Divide both sides of the inequality by the coefficient of x. In this case, divide by -3. However, when dividing or multiplying by a negative number, you must also flip the inequality symbol. So, the inequality becomes x ≥ -1.
Step 4: Graph the inequality x ≥ -1 by drawing a solid line at x = -1 and shading the region to the right of the line.
Step 5: Remember to choose a test point, let's say (0,0), and substitute it into the original inequality -3x + 2 ≤ 5. After simplifying, you should get a true statement, confirming that the shaded region is the solution.
I hope this explanation helps you understand how to graph an inequality and solve problems involving negative coefficients. If you have any further questions, feel free to ask!