Question

consider the inequality -2x + 5 ≤ 7.

1. solve the inequality.
2. graph the solution set.

-5 -4 -3 -2 -1 0 1 2 3 4 5

1. the graphs of y = -2x + 5 and y = 7 are shown on the coordinate plane.

highlight the part of the graph of y = -2x + 5, where the y values are less than or equal to 7.

1. describe the relationship between the solution to the inequality -2x + 5 ≤ 7 and the values of x for which the graph of y = -2x + 5 is less than or equal to 7.

Explanation:

Step1: Isolate the term with x

Subtract 5 from both sides of -2x + 5 ≤ 7.
-2x+5 - 5≤7 - 5, which simplifies to -2x≤2.

Step2: Solve for x

Divide both sides by - 2. When dividing an inequality by a negative number, the inequality sign flips. So $\frac{-2x}{-2}\geq\frac{2}{-2}$, giving x≥ - 1.

Step3: Graph the solution set

On the number - line, draw a closed circle at - 1 (because the inequality is ≥, so - 1 is included in the solution set) and shade to the right of - 1.

Step4: Highlight on the graph of y = - 2x+5

The line y = - 2x + 5 is a straight - line with y - intercept 5 and slope - 2. The line y = 7 is a horizontal line. The part of the graph of y=-2x + 5 where y≤7 is the part of the line y=-2x + 5 that is on or below the line y = 7. This occurs for x≥ - 1.

Step5: Describe the relationship

The solution to the inequality -2x + 5≤7 (x≥ - 1) represents the x - values for which the graph of the linear function y=-2x + 5 has y - values that are less than or equal to 7.

Answer:

1. x≥ - 1
2. Draw a closed circle at - 1 on the number - line and shade to the right.
3. Highlight the part of the line y=-2x + 5 that is on or below the line y = 7.
4. The solution x≥ - 1 gives the x - values for which y=-2x + 5≤7.