Algebra 1: Multi-Step Inequalities & Graphing

Multi-step Inequalities

• Multi-step inequalities require combining like terms and isolating the variable to solve.
• Similar to equations, but the direction of the inequality symbol (>, <, ≥, ≤) must be maintained.
• When multiplying or dividing by a negative number, the inequality symbol must be flipped.

Graphing Inequalities

• Graphing involves first determining the boundary line represented by the inequality.
• For "≤" or "≥" inequalities, the boundary line is solid, indicating that the line’s points are included.
• For "<" or ">" inequalities, the boundary line is dashed, indicating that the line’s points are excluded.
• The solution set is usually shaded toward the direction that satisfies the inequality.

Inequality Examples

• For the inequality ( -4x ≥ -8 ), to solve, divide both sides by -4 (remember to flip the inequality), resulting in ( x ≤ 2 ).
• The notation ( x ≤ 2 ) means any value of x that is less than or equal to 2 will satisfy the inequality.
• For an inequality graph of ( x ≤ 2 ), shade the region to the left of the line ( x = 2 ), including the line itself.

Specific Inequalities

• The inequality ( x ≤ -3 ) indicates that x can be any number less than or equal to -3.
• The inequality ( x ≥ -3 ) indicates that x can be any number greater than or equal to -3.
• The expression ( x - 3 ) suggests a focus on values of x related to the baseline of 3, depending on further context or operations involved.

Inequality Interpretation

• Understanding the graph of inequalities helps visualize solutions and comprehend ranges of values that satisfy the conditions set by the inequalities.
• Each inequality can also have distinct real-world implications, allowing for practical application of these mathematical concepts.