Question

Solve the inequality and represent the solution graphically on number line.

5(2x−7)−3(2x+3)≤0,2x+19≤6x+47

Answer 1

The student's question involves solving two inequalities, graphically representing the solutions on a number line, and finding the overlapping solution. The process involves simplifying each inequality, marking the solution points, and shading the appropriate sections on the number line.

Step-by-step explanation:

To solve the inequalities, let's start with the first one.

1. 5(2x−7)−3(2x+3)≤0 simplifies to 10x − 35 − 6x − 9 ≤ 0, which further simplifies to 4x − 44 ≤ 0. Solving for x gives x ≤ 11.
2. For the inequality 2x+19≤6x+47, simplifying gives −4x≤28, which means x ≥ −7.

To represent the solution graphically on a number line, we draw a number line, mark the points x = 11 and x = −7. Since x ≤ 11, we shade to the left of x = 11 up to negative infinity, which includes x = −7.

We also shade for the second inequality from x = −7 to the right up to positive infinity. Therefore, the overlapping shaded section represents the solution to the system of inequalities, x ≥ −7 and x ≤ 11.

Now, making a number line sketch, we would:

1. Draw a straight horizontal line to represent the number line.
2. Mark points at x = −7 and x = 11.
3. Shade the line segment between x = −7 and x = 11.
4. Put closed circles at x = −7 and x = 11 because the inequalities include the possibility of equality.