Solve and interpret linear inequalities

Which statement is true?

An inequality usually describes many values

This states that x is less than two. Every real value to the left of two satisfies it, including fractions and decimals

Two itself is excluded because the comparison is strict

Less than or equal to includes the boundary two as well as every smaller real value

Greater than points towards larger values; greater than or equal to also includes the boundary

Solving means describing all permitted values, not giving just one example

Inequality

\(\displaystyle x\ge-2\)

Which description is equivalent to the given inequality?

Draw the boundary and extend the correct ray

An open endpoint excludes two. The left ray contains every smaller real value

A closed endpoint includes negative two. The right ray contains every greater real value

An arrow continues indefinitely; the edge of the displayed number line is not a second boundary

Which inequality describes exactly the highlighted real values?

Add, subtract and divide by a positive number

Subtract three from both complete sides. Adding or subtracting the same quantity preserves order

Divide both sides by positive two. A positive divisor preserves order

At the boundary four, the original left side equals eleven, so the strict inequality excludes it

Check one value on each side in the original inequality

These checks support the direction; the equivalent algebraic steps establish the full solution set

Solve a strict two-step inequality

Find all real values of \(x\) that satisfy the inequality. Give the boundary and final comparison symbol separately

Enter the exact boundary value of the solution set

Subtract \(3\) from both sides

Divide both sides by \(2\), which is positive; keep the relation

At the boundary, substitute \(x=4\) into the original expression on each side

Both sides are equal, so the boundary is excluded because the relation is strict

Substitute \(x=3\) into the original inequality

This value lies on the solution side and satisfies the original inequality

Substitute \(x=5\) into the original inequality

This test value is outside the solution set and does not satisfy the original inequality

The open endpoint excludes equality. The ray extends towards smaller values

Choose the relation symbol in the completed solution \(x\) followed by the symbol and your boundary value

Subtract \(3\) from both sides

Divide both sides by \(2\), which is positive; keep the relation

At the boundary, substitute \(x=4\) into the original expression on each side

Both sides are equal, so the boundary is excluded because the relation is strict

Substitute \(x=3\) into the original inequality

This value lies on the solution side and satisfies the original inequality

Substitute \(x=5\) into the original inequality

This test value is outside the solution set and does not satisfy the original inequality

The open endpoint excludes equality. The ray extends towards smaller values

Keep equality while isolating the unknown

A negative multiplier reverses order

Multiplying both numbers in the first comparison by negative two reverses their order. Dividing both sides by a negative number also reverses order

Divide both sides by negative two and reverse less than to greater than

Negative three is included and negative five is excluded, confirming that the ray points right from negative four

Do not reverse the relation merely because a negative number appears, or when adding or subtracting a negative term

Reverse a strict comparison after negative division

Find all real values of \(x\) that satisfy the inequality. Give the boundary and final comparison symbol separately

Enter the exact boundary value of the solution set

Subtract \(3\) from both sides

Divide both sides by \(-2\), which is negative; reverse the relation

At the boundary, substitute \(x=-4\) into the original expression on each side

Both sides are equal, so the boundary is excluded because the relation is strict

Substitute \(x=-3\) into the original inequality

This value lies on the solution side and satisfies the original inequality

Substitute \(x=-5\) into the original inequality

This test value is outside the solution set and does not satisfy the original inequality

The open endpoint excludes equality. The ray extends towards greater values

Choose the relation symbol in the completed solution \(x\) followed by the symbol and your boundary value

Subtract \(3\) from both sides

Divide both sides by \(-2\), which is negative; reverse the relation

At the boundary, substitute \(x=-4\) into the original expression on each side

Both sides are equal, so the boundary is excluded because the relation is strict

Substitute \(x=-3\) into the original inequality

This value lies on the solution side and satisfies the original inequality

Substitute \(x=-5\) into the original inequality

This test value is outside the solution set and does not satisfy the original inequality

The open endpoint excludes equality. The ray extends towards greater values

Reverse direction without losing equality

Which operation requires reversing the inequality symbol?

A boundary may be zero or an exact fraction

Zero is the boundary, not the only solution. Every positive real value satisfies this inequality

Keep the exact fraction. Rounding the boundary could wrongly include or exclude nearby values

Equality is allowed here, so the exact fractional endpoint is included

Describe every value beyond a zero boundary

Retain an exact inclusive fractional boundary

Find all real values of \(x\) that satisfy the inequality. Give the boundary and final comparison symbol separately

Enter the exact boundary value of the solution set

Subtract \(1\) from both sides

Divide both sides by \(3\), which is positive; keep the relation

At the boundary, substitute \(x=\frac{5}{3}\) into the original expression on each side

Both sides are equal, so the boundary is included because equality is allowed

Substitute \(x=0\) into the original inequality

This value lies on the solution side and satisfies the original inequality

Substitute \(x=3\) into the original inequality

This test value is outside the solution set and does not satisfy the original inequality

The closed endpoint includes equality. The ray extends towards smaller values

Choose the relation symbol in the completed solution \(x\) followed by the symbol and your boundary value

Subtract \(1\) from both sides

Divide both sides by \(3\), which is positive; keep the relation

At the boundary, substitute \(x=\frac{5}{3}\) into the original expression on each side

Both sides are equal, so the boundary is included because equality is allowed

Substitute \(x=0\) into the original inequality

This value lies on the solution side and satisfies the original inequality

Substitute \(x=3\) into the original inequality

This test value is outside the solution set and does not satisfy the original inequality

The closed endpoint includes equality. The ray extends towards smaller values

Reverse direction at a negative fractional boundary

Respect every term in a bracket

Expand both terms before collecting, or divide both full sides by the outside factor

The division by negative two reverses the comparison

The original bracket confirms that zero satisfies the inequality and two does not

Expand a negatively scaled bracket

Find all real values of \(x\) that satisfy the inequality. Give the boundary and final comparison symbol separately

Enter the exact boundary value of the solution set

Expand the bracket by multiplying both terms. Dividing both full sides by the outside multiplier is also valid

Subtract \(6\) from both sides

Divide both sides by \(-2\), which is negative; reverse the relation

At the boundary, substitute \(x=1\) into the original expression on each side

Both sides are equal, so the boundary is excluded because the relation is strict

Substitute \(x=0\) into the original inequality

This value lies on the solution side and satisfies the original inequality

Substitute \(x=2\) into the original inequality

This test value is outside the solution set and does not satisfy the original inequality

The open endpoint excludes equality. The ray extends towards smaller values

Choose the relation symbol in the completed solution \(x\) followed by the symbol and your boundary value

Expand the bracket by multiplying both terms. Dividing both full sides by the outside multiplier is also valid

Subtract \(6\) from both sides

Divide both sides by \(-2\), which is negative; reverse the relation

At the boundary, substitute \(x=1\) into the original expression on each side

Both sides are equal, so the boundary is excluded because the relation is strict

Substitute \(x=0\) into the original inequality

This value lies on the solution side and satisfies the original inequality

Substitute \(x=2\) into the original inequality

This test value is outside the solution set and does not satisfy the original inequality

The open endpoint excludes equality. The ray extends towards smaller values

Solve a positive bracket with an included boundary

Clear a positive denominator around the whole numerator

Multiply both complete sides by positive three. The relation is preserved

The fraction bar covers the whole numerator. Equality is allowed, so five is included

Remove a complete numerator denominator

Find all real values of \(x\) that satisfy the inequality. Give the boundary and final comparison symbol separately

Enter the exact boundary value of the solution set

Multiply both full sides by the positive denominator \(3\); keep the relation

Subtract \(-1\) from both sides

Divide both sides by \(2\), which is positive; keep the relation

At the boundary, substitute \(x=5\) into the original expression on each side

Both sides are equal, so the boundary is included because equality is allowed

Substitute \(x=6\) into the original inequality

This value lies on the solution side and satisfies the original inequality

Substitute \(x=4\) into the original inequality

This test value is outside the solution set and does not satisfy the original inequality

The closed endpoint includes equality. The ray extends towards greater values

Choose the relation symbol in the completed solution \(x\) followed by the symbol and your boundary value

Multiply both full sides by the positive denominator \(3\); keep the relation

Subtract \(-1\) from both sides

Divide both sides by \(2\), which is positive; keep the relation

At the boundary, substitute \(x=5\) into the original expression on each side

Both sides are equal, so the boundary is included because equality is allowed

Substitute \(x=6\) into the original inequality

This value lies on the solution side and satisfies the original inequality

Substitute \(x=4\) into the original inequality

This test value is outside the solution set and does not satisfy the original inequality

The closed endpoint includes equality. The ray extends towards greater values

Clear a positive denominator then divide by a negative coefficient

Collect unknown terms without automatically reversing order

Subtract two x from both sides. At each input this subtracts the same quantity, so order is preserved

Both original expressions have value forty-three thirds at the boundary

Alternatively, collecting unknowns on the right creates a negative coefficient. Reverse order only when dividing by that negative coefficient

Find an exact boundary with unknowns on both sides

Find all real values of \(x\) that satisfy the inequality. Give the boundary and final comparison symbol separately

Enter the exact boundary value of the solution set

Subtract the right-hand variable term from both sides

Subtract \(-4\) from both sides

Divide both sides by \(3\), which is positive; keep the relation

At the boundary, substitute \(x=\frac{11}{3}\) into the original expression on each side

Both sides are equal, so the boundary is included because equality is allowed

Substitute \(x=2\) into the original inequality

This value lies on the solution side and satisfies the original inequality

Substitute \(x=5\) into the original inequality

This test value is outside the solution set and does not satisfy the original inequality

The closed endpoint includes equality. The ray extends towards smaller values

Choose the relation symbol in the completed solution \(x\) followed by the symbol and your boundary value

Subtract the right-hand variable term from both sides

Subtract \(-4\) from both sides

Divide both sides by \(3\), which is positive; keep the relation

At the boundary, substitute \(x=\frac{11}{3}\) into the original expression on each side

Both sides are equal, so the boundary is included because equality is allowed

Substitute \(x=2\) into the original inequality

This value lies on the solution side and satisfies the original inequality

Substitute \(x=5\) into the original inequality

This test value is outside the solution set and does not satisfy the original inequality

The closed endpoint includes equality. The ray extends towards smaller values

Collect a negative coefficient and reverse direction

Retain a signed fractional boundary with two unknown terms

Test the original comparison, not only its boundary equation

At the boundary negative two, the original left side equals eight. Strict less than excludes this input

A value to the right satisfies the original comparison

A value to the left does not

The equation for the boundary alone cannot tell you the solution side. Retain the relation throughout the algebra and check it with original values

Testing a few values checks a proposed direction but does not replace the equivalent transformations that establish all solutions

Original inequality

\(\displaystyle -3x+2<8\)

Proposed value

\(\displaystyle x=-2\)

Does the proposed value satisfy the original inequality?

Compare graph heights over a range of inputs

Plot y equal to two x plus one and y equal to five on the same axes

The graphs meet at horizontal coordinate two, which locates the boundary

To the left of the intersection, the rising graph is below the horizontal graph. Its output is less than five

At the intersection the outputs are equal, so the strict inequality excludes x equal to two

The solution is a range of horizontal inputs, not the intersection height five and not only the intersection itself

A plotted reading can be approximate. Use algebra or substitution when exact coordinates are not determined by the graph

Original inequality

\(\displaystyle 2x+1<5\)

Graph intersection

\(\displaystyle P=(2,5)\)

Which solution set follows from comparing the two graph heights?

Condition being tested

\(\displaystyle 2t+1<5\)

Rising graph

\(\displaystyle y=2x+1\)

Change the input t. Point Q lies on the rising graph, while the horizontal line has height five

Which condition on the test input makes the original strict inequality true?

If unknown terms cancel, inspect the constant comparison

The remaining comparison is true independently of x, so every real input is a solution

This remaining comparison is false, so no real input is a solution

Do not divide by a remaining coefficient of zero. Decide whether the constant comparison is true or false

Identical expressions satisfy an inclusive comparison everywhere and a strict comparison nowhere

Inequality

\(\displaystyle -2(x-3)\ge-2x+6\)

What is the solution set of this inequality?

Inequality

\(\displaystyle 4x-1<4x-1\)

What is the solution set of this inequality?

Construct and compare complete solution sets

Independent written and digital task, not automatically marked

Create a linear inequality with a negative coefficient and an exact fractional boundary. Solve it and explain every preservation or reversal of order

Draw its complete number-line solution with a labelled boundary and appropriate endpoint. Check the boundary and a value on each side in the original inequality

Graph both expressions with a digital tool. Explain which horizontal inputs put the first graph above or below the second, and connect that region to your algebraic solution

Change only the strict or inclusive symbol and explain exactly which input is added or removed

Create one cancellation example true for every real input and another with no real solution. Explain both without dividing by zero

Describe all permitted inputs, not just a boundary

Adding or subtracting the same quantity preserves order. Multiplying or dividing by a positive number preserves it, while a negative multiplier or divisor reverses it

Expand every bracket term and clear a whole-numerator denominator from both complete sides

Keep exact boundary values. Retain strict or inclusive status when reversing the relation

Use an open endpoint for a strict relation and a closed endpoint when equality is included. Less than points left; greater than points right

Substitute into the original comparison to check the boundary and test direction

Graphical solutions are the horizontal inputs where the first graph has the required height relationship to the second

If all unknown terms cancel, decide whether the remaining constant comparison is true for every input or false for every input