1. tell whether the ordered pair is a solution of the inequality. + y

Question

1. tell whether the ordered pair is a solution of the inequality. + y < 7; (2,3) 2. graph the inequality: y ≤ 5 3. graph the inequality: y ≥ -x - 1 4. describe and correct the error in graphing the inequality: y < -x + 1 5. write the inequality that represents the graph. 6. you have at most $25 to spend on in - app purchases for a game. hints cost $5 each, and energy boosters cost $1 each. write and graph an inequality that represents the numbers of each item you can buy. identify and interpret two solutions of the inequality. 7. tell whether the ordered pair is a solution of the inequality. x + 3y ≥ -2; (-9,1) 8. graph the inequality. x < 2 9. graph the inequality: 5x - 2y ≤ 6

Accepted Answer

### Problem 1: (Assuming the first inequality is $ x + y < 7 $) # Explanation: ## Step 1: Substitute the ordered pair Substitute $ x = 2 $ and $ y = 3 $ into the inequality $ x + y < 7 $. $ 2 + 3 < 7 $ ## Step 2: Simplify and check Simplify the left - hand side: $ 2+3 = 5 $. Now we check if $ 5 < 7 $, which is true. # Answer: The ordered pair $(2,3)$ is a solution of the inequality. ### Problem 2: Graph $ y\leq5 $ # Explanation: ## Step 1: Identify the boundary line The inequality $ y\leq5 $ has a boundary line $ y = 5 $. Since the inequality is "less than or equal to", the boundary line is a horizontal line (parallel to the $ x $ - axis) and it is a solid line (because the inequality includes equality). ## Step 2: Determine the region to shade We test a point not on the line, for example, the origin $(0,0)$. Substitute $ y = 0 $ into the inequality: $ 0\leq5 $, which is true. So we shade the region below the line $ y = 5 $ (including the line itself because of the "equal to" part). # Answer: The graph is a solid horizontal line $ y = 5 $ with the region below (including the line) shaded. ### Problem 3: Graph $ y\geq - x - 1 $ # Explanation: ## Step 1: Identify the boundary line The boundary line is $ y=-x - 1 $. The slope of the line $ m=-1 $ and the $ y $ - intercept $ b=-1 $. Since the inequality is "greater than or equal to", the boundary line is a solid line. ## Step 2: Determine the region to shade Test the origin $(0,0)$: Substitute $ x = 0 $ and $ y = 0 $ into $ y\geq - x - 1 $, we get $ 0\geq - 0 - 1$, i.e., $ 0\geq - 1 $, which is true. So we shade the region above the line $ y=-x - 1 $ (including the line itself). # Answer: The graph is a solid line with slope - 1 and $ y $ - intercept - 1, and the region above (including the line) is shaded. ### Problem 4: (Error in graphing $ y < -x + 1 $) # Explanation: ## Step 1: Identify the correct boundary line and shading The inequality $ y < -x + 1 $ has a boundary line $ y=-x + 1 $ (slope $ m=-1 $, $ y $ - intercept $ b = 1 $). Since the inequality is "less than", the boundary line should be a dashed line (not solid) because the points on the line do not satisfy $ y < -x + 1 $. Also, to determine the shading, test the origin $(0,0)$: $ 0<-0 + 1$ (i.e., $ 0 < 1 $) which is true, so we should shade the region below the line $ y=-x + 1 $. If the original graph had a solid line or shaded the wrong region, we correct it: use a dashed line for $ y=-x + 1 $ and shade the region where $ y < -x + 1 $ (below the line). # Answer: Error: The boundary line may be solid (it should be dashed for $ y < -x + 1 $) or the shading region is incorrect. Correction: Draw a dashed line $ y=-x + 1 $ and shade the region below the line (where $ y < -x + 1 $). ### Problem 5: Write the inequality for the graph (dashed line with slope 1, $ y $ - intercept 1) # Explanation: ## Step 1: Identify the boundary line equation The line has a slope $ m = 1 $ and $ y $ - intercept $ b = 1 $, so the equation of the line is $ y=x + 1 $. Since the line is dashed, the inequality is either $ y>x + 1 $ or $ yx + 1 $: $ 2>0 + 1$ (i.e., $ 2 > 1 $) which is true. So the inequality is $ y>x + 1 $. # Answer: The inequality is $ y>x + 1 $. ### Problem 6: # Explanation: ## Step 1: Define variables Let $ x $ be the number of hints and $ y $ be the number of energy boosters. ## Step 2: Write the inequality The cost of hints is $ 5x $ dollars and the cost of energy boosters is $ 1y=y $ dollars. The total cost must be at most $ 25 $ dollars. So the inequality is $ 5x + y\leq25 $, where $ x\geq0 $ and $ y\geq0 $ (since the number of items can't be negative). ## Step 3: Find two solutions - Solution 1: Let $ x = 0 $, then $ y\leq25 $. Take $ y = 25 $. This means we can buy 0 hints and 25 energy boosters. - Solution 2: Let $ x = 1 $, then $ 5(1)+y\leq25\Rightarrow y\leq20 $. Take $ y = 20 $. This means we can buy 1 hint and 20 energy boosters. ## Step 4: Graph the inequality The boundary line is $ 5x + y=25 $ (rewrite as $ y=-5x + 25 $), slope $ m=-5 $, $ y $ - intercept $ b = 25 $, $ x $ - intercept (when $ y = 0 $) $ x = 5 $. The line is solid (because of $ \leq $). Test the origin $(0,0)$: $ 5(0)+0\leq25$ (i.e., $ 0\leq25 $) which is true, so shade the region below the line $ y=-5x + 25 $ with $ x\geq0 $ and $ y\geq0 $. # Answer: Inequality: $ 5x + y\leq25 $, $ x\geq0 $, $ y\geq0 $. Solution 1: $(0,25)$ (0 hints, 25 energy boosters). Solution 2: $(1,20)$ (1 hint, 20 energy boosters). Graph: Solid line $ y=-5x + 25 $, shade below the line in the first quadrant. ### Problem 7: Check if $(-9,1)$ is a solution of $ x + 3y\geq - 2 $ # Explanation: ## Step 1: Substitute the values Substitute $ x=-9 $ and $ y = 1 $ into the inequality $ x + 3y\geq - 2 $. $-9+3(1)\geq - 2$ ## Step 2: Simplify and check Simplify the left - hand side: $-9 + 3=-6$. Now check if $-6\geq - 2$. Since $-6<-2$, the inequality is not satisfied. # Answer: The ordered pair $(-9,1)$ is not a solution of the inequality $ x + 3y\geq - 2 $. ### Problem 8: Graph $ x < 2 $ # Explanation: ## Step 1: Identify the boundary line The boundary line is $ x = 2 $, which is a vertical line (parallel to the $ y $ - axis). Since the inequality is "less than", the boundary line is dashed (points on $ x = 2 $ do not satisfy $ x < 2 $). ## Step 2: Determine the region to shade All points with $ x $ - coordinate less than 2 satisfy the inequality. So we shade the region to the left of the dashed line $ x = 2 $. # Answer: Graph: Dashed vertical line $ x = 2 $, shade the region to the left of the line. ### Problem 9: Graph $ 5x-2y\leq6 $ # Explanation: ## Step 1: Rewrite the inequality in slope - intercept form Solve $ 5x-2y\leq6 $ for $ y $: $-2y\leq - 5x + 6$ Divide both sides by $-2$ (and reverse the inequality sign): $ y\geq\frac{5}{2}x - 3 $ ## Step 2: Identify the boundary line The boundary line is $ y=\frac{5}{2}x - 3 $ (slope $ m=\frac{5}{2} $, $ y $ - intercept $ b=-3 $). Since the inequality is "greater than or equal to", the boundary line is solid. ## Step 3: Determine the shading region Test the origin $(0,0)$: Substitute into $ y\geq\frac{5}{2}x - 3 $, $ 0\geq\frac{5}{2}(0)-3$ (i.e., $ 0\geq - 3 $) which is true. So shade the region above the line $ y=\frac{5}{2}x - 3 $. # Answer: Graph: Solid line $ y=\frac{5}{2}x - 3 $, shade the region above the line.

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QUESTION IMAGE

Question

Explanation:

Problem 1: (Assuming the first inequality is \( x + y < 7 \))

Step 1: Substitute the ordered pair

Step 2: Simplify and check

Step 1: Identify the boundary line

Step 2: Determine the region to shade

Step 1: Identify the boundary line

Step 2: Determine the region to shade

Answer:

Problem 2: Graph \( y\leq5 \)