Question

solving systems of inequalities:

• make sure both inequalities are in ____________.
• graph the __________ inequality and shade.
• graph the ______________ inequality and shade.
• find the __________ in which they __________.

Explanation:

To solve a system of inequalities, the standard steps are:

1. First, ensure both inequalities are in slope - intercept form (i.e., \(y = mx + b\) form, or for inequalities, \(y\gt mx + b\), \(y\lt mx + b\), \(y\geq mx + b\), \(y\leq mx + b\)). This is because the slope - intercept form makes it easy to identify the slope and the y - intercept, which are crucial for graphing the line and determining the region to shade.
2. Then, graph the first inequality. When graphing, if the inequality is strict (e.g., \(y\gt\) or \(y\lt\)), we use a dashed line, and if it is non - strict (e.g., \(y\geq\) or \(y\leq\)), we use a solid line. After drawing the line, we shade the region that satisfies the inequality.
3. Next, graph the second inequality in the same coordinate plane. Again, we use the appropriate line style (dashed or solid) and shade the region that satisfies this second inequality.
4. Finally, find the region in which they overlap (intersect). The overlapping region is the solution to the system of inequalities, as it represents the set of points that satisfy both inequalities simultaneously.

Answer:

• Make sure both inequalities are in \(\boldsymbol{\text{slope - intercept form}}\).
• Graph the \(\boldsymbol{\text{first}}\) inequality and shade.
• Graph the \(\boldsymbol{\text{second}}\) inequality and shade.
• Find the \(\boldsymbol{\text{region}}\) in which they \(\boldsymbol{\text{overlap}}\).